Date: Mon, 20 Jun 1994 18:43:23 +0500 (GMT+4:00)
From: categories
Subject: Topology: a geometric account of general topology, homotopy types and t
he fundamental groupoid, published 1988, by Ronald Brown
Date: Mon, 20 Jun 1994 14:28:33 +0100
From: "Prof R. Brown"
The publishers have just informed informed me that the above book has been
out of print since sept, 1993, that no stock is available, and that the
copyright reverts to me.
I would consider getting some loose page copies printed and sold if this
seemed economic, and/or if for example there was a desire to use part or all
for course adoption. For the latter I could supply a top copy and allow for
copying to students at a moderate fee per student. Offers welcomed. The aim
is to get the book available and read. The publisher's previous distribution
at 50 pounds sterling discouraged this.
This book has material on groupoids, homotopy types, fundamental groupoid of
orbit spaces, etc. not available elsewhere.
Ronnie Brown
Prof R. Brown Tel: +44 248 382474
School of Mathematics Fax: +44 248 355881
Dean St email: mas010@uk.ac.bangor
University of Wales
Bangor
Gwynedd LL57 1UT
UK
Date: Fri, 17 Jun 1994 13:59:52 +0500 (GMT+4:00)
From: categories
Subject: ANNOUNCE: KockMoerdijk preprint
Date: Fri, 17 Jun 1994 08:40:31 +0200
From: kock@mi.aau.dk
Preprint "Spaces with local equivalence relations, and their
monodromy" available.
The final version of Kock and Moerdijk's work on this topic is
now available by ftp at theory.doc.ic.ac.uk (login: anonymous,
directory: papers/Kock, file: kmler.dvi). Compared to the 1991
preprint, it has more emphasis on topology, less on toposes
(the topos theoretic aspects are dealt with in our article
published in JPAA 82 (1992)).
A paper version (Aarhus Preprint 1994 No. 8) is also available,
on request.
Date: Wed, 22 Jun 1994 22:36:48 +0500 (GMT+4:00)
From: categories
Subject: in case you hadn't heard
Date: Tue, 21 Jun 94 11:51:50 0400
From: jds@math.upenn.edu
available on the hepth list server is a paper by
V. Lychagin
Calculus and quantizations over Hopf algebras
hepth 9406097
It includes ``a general notion of quantization in monoidal cats''
which `deforms all natural algebraic and differential objects''.
This occurs in Section 3 p. 31 + so skim rapidly if this is the
part that interests you.
[Note from moderator: th poster provided the following information on
accessing hepth server]
This bulletin board for string/conformal field theory/2d gravity preprints,
hepth@xxx.lanl.gov, described in an earlier message is now turned on.
For the first week or so, replies to messages received during the day will
be sent out only during the evening (so that potential bugs can be
identified and corrected under actual combat conditions).
A (subscribe all / cancel all) option has been added that allows automatic
receipt of the full text of each paper on day of receipt.
This is recommended only for large groups where a single such account could
be used for general distribution or posting in a preprint library.
Commands to the system, entered in the Subject: field of messages, are
get # returns to requester the paper specified by #
put submit paper (body of message in format described below*)
paper is assigned #, and added to listing
replace # replace paper specified by # with revised version (only
original submitter can do this).
listing returns title/author list of all papers currently held
find (keyword) search title/author list for keyword (either authorname or
word in title) to recall #
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automatically extracted from return address)
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subscribe all automatically receive full text of all papers
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help returns this list of commands
Date: Thu, 23 Jun 1994 14:21:29 +0500 (GMT+4:00)
From: categories
Subject: Domain Theory and Partial Maps
Date: Thu, 23 Jun 1994 17:47:33 +0000
From: Marcelo Fiore
My thesis (Axiomatic Domain Theory in Categories of Partial Maps) is
available by anonymous ftp from ftp.dcs.ed.ac.uk directory pub/mf files
thesis.dvi.Z and thesis.ps.Z.
Marcelo Fiore
P.S. The abstract follows:

Axiomatic Domain Theory
in Categories of Partial Maps
Marcelo P. Fiore
Department of Computer Science
Laboratory for Foundations of Computer Science
University of Edinburgh, The King's Buildings
Edinburgh EH9 3JZ, Scotland
Phone: + 44 31 650 5145.
Fax: + 44 31 667 7209.
June 1994
Synopsis
This thesis is an investigation into axiomatic categorical domain
theory as needed for the denotational semantics of deterministic
programming languages.
To provide a direct semantic treatment of nonterminating computations,
we make partiality the core of our theory. Thus, we focus on categories
of partial maps. We study representability of partial maps and show its
equivalence with classifiability. We observe that, once partiality is
taken as primitive, a notion of approximation may be derived. In fact,
two notions of approximation, contextual approximation and
specialisation, based on testing and observing partial maps are
considered and shown to coincide. Further we characterise when the
approximation relation between partial maps is domaintheoretic in the
(technical) sense that the category of partial maps Cpoenriches with
respect to it.
Concerning the semantics of type constructors in categories of partial
maps, we present a characterisation of colimits of diagrams of total
maps; study orderenriched partial cartesian closure; and provide
conditions to guarantee the existence of the limits needed to solve
recursive type equations. Concerning the semantics of recursive types,
we motivate the study of enriched algebraic compactness and make it the
central concept when interpreting recursive types. We establish the
fundamental property of algebraically compact categories, namely that
recursive types on them admit canonical interpretations, and show that
in algebraically compact categories recursive types reduce to inductive
types. Special attention is paid to Cpoalgebraic compactness, leading
to the identification of a 2category of kinds with very strong closure
properties.
As an application of the theory developed, enriched categorical models
of the metalanguage FPC (a type theory with sums, products,
exponentials and recursive types) are defined and two abstract examples
of models, including domaintheoretic models, are axiomatised. Further,
FPC is considered as a programming language with a callbyvalue
operational semantics and a denotational semantics defined on top of a
categorical model. Operational and denotational semantics are related
via a computational soundness result. The interpretation of FPC
expressions in domaintheoretic Posetmodels is observed to be
representationindependent. And, to culminate, a computational adequacy
result for an axiomatisation of absolute nontrivial domaintheoretic
models is proved.

Date: Thu, 7 Jul 1994 11:24:29 +0500 (GMT+4:00)
From: categories
Subject: Revised version of Acyclic models
Date: Thu, 7 Jul 94 09:02:39 EDT
From: Michael Barr
A totally revised version of Acyclic models is now available for ftp
on my directory on triples. .tex, .dvi and .ps versions are there.
Michael
Date: Thu, 7 Jul 1994 21:53:53 +0500 (GMT+4:00)
From: categories
Subject: Re: Empty types and typed lambda calculus
To: categories
MessageId:
MimeVersion: 1.0
ContentType: TEXT/PLAIN; charset=USASCII
Status: RO
Date: Thu, 07 Jul 94 09:37:26 0700
From: "John C. Mitchell"
We looked into this kind of thing about seven or eight years ago. See
@inproceedings(
mmms87, author="A. R. Meyer and J. C. Mitchell and E. Moggi and R. Statman",
Key="MMMS 87",
Title="Empty types in polymorphic lambda calculus",
Booktitle="Proc. 14th ACM Symp. on Principles of
Programming Languages",
Month="January",Year="1987",
pages="253262",
Note="Reprinted with minor revisions in
{\it Logical Foundations of Functional Programming,}
ed. G. Huet, AddisonWesley (1990) 273284.")
@article(
mm87, author="Mitchell, J.C. and Moggi, E.",
Title="Kripkestyle models for typed lambda calculus",
Journal="Ann. Pure and Applied Logic",
Volume="51",Year="1991",
pages="99124",
Note="Preliminary version in
{\it Proc. IEEE Symp. on Logic in Computer Science,} 1987, pages
303314.")
@incollection(
MitchScott,Author="Mitchell, J.C. and Scott, P.J.",
Title="Typed lambda calculus and cartesian closed
categories",
Booktitle="Categories in Computer Science and
Logic, Proc. Summer Research Conference,
Boulder, Colorado, June, 1987",
Series="Contemporary Mathematics",
volume="92",
publisher="Amer. Math. Society",
editors="J.W. Gray and A. Scedrov",
Year="1989",
pages="301316")
Friedman's completeness theorem is completeness of beta,eta for
full classical model (i.e., Set). Deductive completeness fails
for Set and for models (interpretations into categories) where
every type (every object named by a type expression) is required
to be nonempty (have a global element). I tried to clarify this
in my book (still to appear). If anyone is seriously interested,
I could email the appropriate sections of the book.
John Mitchell
te: Tue, 19 Jul 1994 13:38:12 +0500 (GMT+4:00)
From: categories
Subject: Re: Anouncement of Preprints
Date: Tue, 19 Jul 1994 16:33:59 +1000
From: Murray Adelman
I have mounted four papers by John Corbett and myself that attempt to
use sheaf theory to interpret Quantum Logic (or our version thereof)
on the Sydney University ftp site.
They are called
AdelmanCorbett.ps Long with a lot of expository material for
physicists.
comparison.psAttempts to compare Birkhoff and vonNeumann Quantum
logic to the internal logic of the category of sheaves over state
space
newpaper.ps Attempts to show how continuous data can become
discrete data via the global sections functor
heidelberg.psDerives a formula for interference of particles that
allows more of the configuration of the experiment to be parametrized
than the usual Hilbert Space description.
The union of heidelberg.ps and comparison.ps approximates (and perhaps
supersedes) AdelmanCorbett.ps which is somewhat older.
They can be obtained by anonymous ftp from
maths.su.oz.au
in the directory
sydcat/papers/murray
Regards,
Murray
Date: Tue, 26 Jul 1994 09:09:33 +0500 (GMT+4:00)
From: categories
Subject: New edition of "Category Theory for Computing Science"
Date: Mon, 25 Jul 1994 14:13:54 0400
From: "Charles F. Wells"
We are revising our text, "Category Theory for Computing
Science", for its second edition. We would appreciate any
comments or suggestions concerning this revision, including
papers and books we should refer to, new topics we should
include, and topics we could delete. Please reply to either of
us.
Michael Barr Charles Wells
barr@triples.math.mcgill.ca cfw2@po.cwru.edu

Charles Wells, Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland OH 441067058, USA
Phone 216 368 2880 or 216 774 1926
FAX 216 368 5163
Date: Tue, 26 Jul 1994 09:14:44 +0500 (GMT+4:00)
From: categories
Subject: Papers
Date: Sat, 23 Jul 1994 11:53:22 +1000
From: Murray Adelman
There seems to be some problem obtaining the papers that I announced a
few days ago. I have put them in a second location
ftp.mpce.mq.edu.au
in the directory
/pub/maths/murray
(This is more patriotic, as it is the Macquarie University server
instead of the Sydney University server :)
Sorry for the inconvenience.
Murray
From schreine Mon Aug 29 20:54 MDT 1994
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Date: Mon, 29 Aug 1994 15:34:59 +0500 (GMT+4:00)
From: categories
Subject: Top\op is a quasivariety
To: categories
MessageId:
MimeVersion: 1.0
ContentType: TEXT/PLAIN; charset=USASCII
Status: RO
Date: Sat, 27 Aug 94 14:47:11 EDT
From: Michael Barr
by Michael Barr and M. Cristina Pedicchio
We show that there is a certain variety (= category tripleable over
sets) and a simple Horn sentence in it of the form phi(u) = phi(v) ==>
psi(u) = psi(v) whose category of models is equivalent to the opposite
of topological spaces. The theory consists of that of frames together
with a unary operation we denote ' (it is a kind of pseudocomplement)
satisfying a small set of equations plus an equation scheme that forces
all intervals of the form [u /\ u',u \/ u'] to be complete atomic
boolean algebras with the Sup and ' as operations. The underlying set
functor on Top\op takes a space to the set of all pairs (U,A) where U is
open and A is an arbitrary subset of U. The frame operations are the
usual, while (U,A)' = (U,U  A). The Horn clause is u \/ u' \/ 1' = v
\/ v' \/ 1' ==> u \/ u' = v \/ v'.
[note from moderator: Michael says the paper will be available by ftp from
triples.math.mcgill.ca soon.]
Date: Sat, 10 Sep 1994 22:54:25 +0500 (GMT+4:00)
From: categories
Subject: "Homotopical algebra and triangulated categories"
Date: Thu, 8 Sep 1994 09:35:25 +0200
From: Marco Grandis
The following preprint is available (by hard mail)
Marco Grandis, "Homotopical algebra and triangulated categories"
to appear in: Math. Proc. Cambridge Philos. Soc.
Abstract
We study here the connections between the well known PuppeVerdier
notion of triangulated category and an abstract setting for homotopical
algebra, based on homotopy kernels and cokernels, which was exposed by the
author in two previous papers ["On the categorical foundations of
homological and homotopical algebra", Cahiers Top. Geom. Diff. Categ. 33
(1992), 135175. "Homotopical algebra in homotopical categories", Appl.
Categ. Struct., to appear].
We show that a righthomotopical category A (having wellbehaved
homotopy cokernels, i.e. mapping cones) has a sort of weak triangulated
structure with regard to the suspension endofunctor Sigma, called
Sigmahomotopical category. If A is right and lefthomotopical and
hstable (in a sense related to the suspensionloop adjunction), also this
structure is hstable, i.e. satisfies "up to homotopy" the axioms of
Verdier for a triangulated category, excepting the octahedral one which
depends on some further elementary conditions on the cone endofunctor of A.
Every Sigmahomotopical category can be stabilised, by two
universal procedures, respectively initial and terminal.
Date: Tue, 11 Oct 1994 08:59:49 +0500 (GMT+4:00)
From: categories
Subject: Graphbased Logic and Sketches
Date: 7 Oct 1994 19:30:41 GMT
From: Charles Wells
"Graphbased Logic and Sketches I: The General Framework" by
Atish Bagchi and Charles Wells is now available by anonymous FTP
from ftp.cwru.edu in the directory math/wells. It is also
available by gopher at gopher.cwru.edu, path
1/class/mans/math/pub/wells.
In either case the files are logstr.dvi and logstr.ps. To print
the dvi file requires AMS fraktur and the 1992 version of the
xypic fonts.

Charles Wells
Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland, OH 441067058, USA
216 368 2893
Date: Fri, 14 Oct 1994 18:57:05 +0500 (GMT+4:00)
From: categories
Subject: Lambda Definability in Categorical Models
Date: Tue, 11 Oct 1994 18:02:14 0400
From: Andre Scedrov
A Characterization of Lambda Definability
in Categorical Models of Implicit Polymorphism
Moez Alimohamed
University of Pennsylvania
This paper contains the work of Moez Alimohamed, a mathematics graduate
student at the University of Pennsylvania who died tragically on August 29th.
Lambda definability is characterized in categorical models of simply typed
lambda calculus with type variables. A categorytheoretic framework known
as glueing or sconing is used to extend the JungTiuryn characterization of
lambda definability in Henkin models for the simply typed lambda calculus
first to ccc models, and then to categorical models of the calculus with
type variables.
WWW access is http://www.cis.upenn.edu/~andre/moez.html. The paper is also
available by anonymous ftp from the host ftp.cis.upenn.edu as the file
pub/papers/scedrov/def.ps.Z.
te: Wed, 9 Nov 1994 15:53:25 +0400 (GMT+4:00)
From: categories
Subject: New papers available by gopher & ftp
Date: 8 Nov 1994 20:51:40 GMT
From: Charles Wells
Three papers are newly available by gopher and by anonymous ftp
from Case Western Reserve University:
"The categorical theory generated by a limit sketch", by Michael
Barr and Charles Wells. This was formerly called "The category
of diagrams".
"Varieties of mathematical prose" by Atish Bagchi and Charles
Wells. This is not about category theory but was referred to in
our paper "The logic of sketches" (also available by gopher &
ftp).
"Extension theories for categories", by Charles Wells. This is
an old paper that was never submitted for publication. I have updated
the bibliography as far as I can, but would appreciate it if
anyone knows of other suitable references.
They are available by gopher in both DVI and Postscript form
from the host gopher.cwru.edu, path 1/class/mans/math/pub/wells.
Most www clients can get gopher files, and some of the can view
Postscript files on screen.
The files are also available by anonymous ftp from ftp.cwru.edu in
the directory math/wells, under the following filenames:
"Varieties of mathematical prose": MATHRITE.DVI, MATHRITE.PS
"Extension theories for categories": CATEXT.DVI, CATEXT.PS
"The categorical theory generated by a limit sketch": DIAGC.DVI,
DIAGC.PS
"The logic of sketches": LOGSTR.DVI, LOGSTR.PS
I would appreciate knowing if anyone has problems downloading
these files.

Charles Wells
Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland, OH 441067058, USA
216 368 2893
Date: Thu, 10 Nov 1994 18:47:37 +0400 (GMT+4:00)
From: categories
Subject: Announce paper: Cockett & Seely
Date: Wed, 9 Nov 94 22:38:25 EST
From: "Robert A. G. Seely"
The following paper has been placed on anonymous ftp at
triples.math.mcgill.ca, directory /pub/rags/wk_dist_cat
Weakly distributive categories
by J.R.B. Cockett and R.A.G. Seely
Abstract:
There are many situations in logic, theoretical computer science, and
category theory where two binary operationsone thought of as a (tensor)
``product'', the other a ``sum''play a key role. In distributive and
*autonomous categories these operations can be regarded as, respectively,
the and/or of traditional logic and the times/par of (multiplicative) linear
logic. In the latter logic, however, the distributivity of product over sum
is conspicuously absent: this paper studies a ``linearization'' of that
distributivity which is present in both case. Furthermore, we show that
this weak distributivity is precisely what is needed to model Gentzen's cut
rule (in the absence of other structural rules) and can be strengthened in
two natural ways to generate full distributivity and *autonomous categories.
This is the journal version of the similarly named paper appearing in the
Proceedings of the Durham conference (1991):
M.P. Fourman, P.T. Johnstone, A.M. Pitts, eds., Applications of Categories
to Computer Science, London Mathematical Society Lecture Note Series 177
(1992) 45  65.
This version is to appear in the Journal of Pure and Applied Algebra.
The paper has been rewritten, including more details and several examples,
including shifted tensors, Span categories, and categories of modules of a
bialgebra.
Date: Thu, 10 Nov 1994 18:50:26 +0400 (GMT+4:00)
From: categories
Subject: Announce paper: Mendler, Panangaden, Scott, & Seely
Date: Wed, 9 Nov 94 22:40:42 EST
From: "Robert A. G. Seely"
The following paper has been placed on anonymous ftp at
triples.math.mcgill.ca in directory (file) /pub/rags/ccp/mpss.*
A Logical View of Concurrent Constraint Programming
by
Nax Paul Mendler
Prakash Panangaden
P.J. Scott
R.A.G. Seely
Abstract;
The Concurrent Constraint Programming paradigm has been the subject of
growing interest as the focus of a new paradigm for concurrent computation.
Like logic programming it claims close relations to logic. In fact these
languages _are_ logics in a certain sense that we make precise in this
paper. In recent work it was shown that the denotational semantics of
determinate concurrent constraint programming languages forms a categorical
structure called a hyperdoctrine, which is used as the basis of the
categorical formulation of first order logic. What this connection shows
is the combinators of determinate concurrent constraint programming can be
viewed as logical connectives. In the present work we extend these ideas
to the operational semantics of these languages and thus make available
similar analogies for a much broader variety of languages including the
indeterminate concurrent constraint programming languages and concurrent
blockstructured imperative languages.
\begin{BTW}
In the same directory you will find the earlier paper dealing with the
denotational semantics (psss*). This paper has appeared:
P. Panangaden, V. Saraswat, P. J. Scott, R. A. G. Seely,
A Hyperdoctrinal View of Concurrent Constraint Programming, in J.W. de Bakkee
et al, eds. Semantics: Foundations and Applications; Proceedings of
REX Workshop, Beekbergen, The Netherlands, June 1992. Springer Lecture
Notes in Comp. Science, 666 (1993) pp. 457  476.
\end{BTW}
Date: Thu, 10 Nov 1994 13:01:16 +0400 (GMT+4:00)
From: categories
Subject: cat logic of interactions
Date: Wed, 9 Nov 1994 22:10:43 +0000 (GMT)
From: Dusko Pavlovic
I just remembered that I never announced this paper in which I was
trying to learn some concurrency theory. It depends on a note on
representation, that used to be its appendix. It will be available
soon.
The psfile with A4format is in papers/Pavlovic on
theory.doc.ic.ac.uk; the American format is in pub/pavlovic on
triples.math.mcgill.ca.
All the best,
 Dusko
CATEGORICAL LOGIC OF CONCURRENCY AND INTERACTION I:
SYNCHRONOUS PROCESSES
by Dusko Pavlovic (August 1994)
Abstract.
This is a report on a mathematician's effort to understand some
concurrency theory. The starting point is a logical interpretation of
Nielsen and Winskel's account of the basic models of concurrency. Upon
the obtained logical structures, we build a calculus of relations
which yields, when cut down by bisimulations, Abramsky's interaction
category of synchronous processes. It seems that all interaction
categories arise in this way. The obtained presentation uncovers some
of their logical contents and perhaps sheds some more light on the
original idea of processes as relations extended in time.
The sequel of this paper will address the issues of asynchronicity,
preemption, noninterleaving and linear logic in the same setting.
Date: Tue, 15 Nov 1994 19:29:48 0400 (AST)
From: categories
To: categories
Subject: Announcement
Date: Tue, 15 Nov 94 15:27:16 EST
From: Michael Makkai
This is to announce two papers,
"Avoiding the axiom of choice in general category theory"
and
"Generalized sketches as a framework for completeness theorems"
both by M. Makkai, McGill University.
Both papers are revised versions of ones with identical titles; the
original versions were produced about a year ago. Both papers will
appear in the Journal of Pure and Applied Algebra. Unfortunately, they
are not available electronically. If you are interested in obtaining
copies, please send your request to Makkai@triples.math.mcgill.ca .
The abstracts of the papers follow.
ABSTRACT of "Avoiding the axiom of choice in general category theory",
by M. Makkai, McGill University:
"The notion of anafunctor is introduced. An anafunctor is, roughly, a
"functor defined up to isomorphism". Anafunctors have a general
theory paralleling that of ordinary functors; they have natural
transformations, they form categories, they can be composed, etc.
Anafunctors can be saturated, to ensure that any object isomorphic to
a possible value of the anafunctor is also a possible value at the
same argument object. The existence of anafunctors in situations when
ordinarily one would use choice is ensured without choice; e.g., for a
category which has binary products, but not specified binary products,
the anaversion of the product functor is canonically definable, unlike
the ordinary product functor that needs the axiom of choice. When the
composition functors in a bicategory are changed into anafunctors, one
obtains anabicategories. In the standard definitions of bicategories
such as the monoidal category of modules over a ring, or the
bicategory of spans in a category with pullbacks, and many others, one
uses choice; the anaversions of these bicategories have canonical
definitions. The overall effect is an elimination of the axiom of
choice, and of noncanonical choices, in large parts of general
category theory. To ensure the Cartesian closed character of the
bicategory of small categories, with anafunctors as 1cells, one uses
a weak version of the axiom of choice, which is related to A. Blass'
axiom of Small Violations of Choice ("Injectivity, projectivity, and
the axiom of choice", Trans. Amer. Math. Soc. 255(1979), 3159)."
ABSTRACT of "Generalized sketches as a framework for completeness
theorems", by M. Makkai, McGill University:
"A generalized concept of sketch is introduced. Because of their role,
morphisms of (generalized) sketches are called sketchentailments. A
sketch is said to satisfy a sketchentailment if the former is
injective relative to the latter in the standard sense; the models of
a set R of sketchentailments are the sketches satisfying all members
of R . R logically implies a sketchentailment s if every model of R
is also a model of {s} . A deductive calculus is introduced in which
s is deducible from R iff R logically implies s (General Completeness
Theorem, GCT). A large number of examples of kinds of structured
category is presented showing that the structured categories are
selected from among the corresponding generalized sketches as the
models of a set of sketchentailments. As a consequence, the
sketchentailments satisfied by all structured categories of a given
kind are exactly the ones that are deducible from a certain, usually
finite, set of axioms. In the finitary case, which is the only one
considered in detail in the paper, the notion of deduction is
effective, and straightforwardly implementable on a computer. One
obtains Specific Completeness Theorems (SCT's), each of which asserts
that the exactness properties (certain kinds of sketchentailments)
that hold in a specific class of structured categories coincide with
the ones that are deducible from a given set of axioms. Each of these
specific completeness theorems is deduced from the GCT, and a
particular Representation Theorem (RT); RT's are a wellknown class of
results in categorical logic. The concepts of Compactness and of
Abstract Completeness are introduced, and shown to correspond to the
samenamed concepts in logics in the usual symbolic form, via a
translation between the sketchbased syntax and semantics on the one
hand, and the Tarskian syntax and semantics on the other. The
sketchbased concepts are available for several logics defined
categorically for which there are no available symbolic
presentations."
Date: Wed, 16 Nov 1994 18:47:19 0400 (AST)
From: categories
To: categories
Subject: Availability of new paper by ftp
Date: Wed, 16 Nov 94 16:56:03 +1100
From: Max Kelly
A new preprint "On localization and stabilization for factorization
systems", by Carboni, Janelidze, Kelly, and Pare', is available in our ftp
site at the address maths.su.oz.au (= 129.78.68.2), in the directory
sydcat/papers/kelly, under the titles cjkp.dvi or cjkp.ps; there is also
cjkp.tex, but that requires two macros  namely diagrams.tex and
kluwer.sty.
The paper contains new ideas, but also selfcontained introductions to
several areas with which some may be unfamiliar: namely factorization
systems, descent theory, Galois theory, Eilenberg's monotonelight
factorization for maps between compact hausdorff spaces, hereditary
torsion theories for abelian categories, the category of finite families
of objects of a given category, and the (separable, purelyinseparable)
factorization for field extensions.
What ways are there of constructing a factorization system (E, M) on a
category C ? One simple one is to start with a full reflective subcategory
X of C, and to take E to consist of the maps inverted by the reflexion. Of
course, this (E, M) doesn't have E pullbackstable except in the special
case where X is a LOCALIZATION of C.
We examine another general process, which leads to an (E, M) with E stable
when it succeeds. We start from ANY factorization system (E, M), and
define new classes thus: a map lies in E' if EACH of its pullbacks lies in
E; and it lies in M* if SOME pullback of it along an effective descent map
lies in M. Note the connexion with Galois theory, in Janelidze's
categorical formulation of it: if the (E, M) we begin with arises as above
from a reflective full subcategory, the class M* consists of what
Janelidze calls the COVERINGS (or, in some contexts, the CENTRAL
EXTENSIONS).
It is not always the case that (E', M*) is a factorization system; we give
necessary and sufficient conditions for it to be so, and apply these to
three major examples: Eilenberg's factorization above, certain
factorizations connected to hereditary torsion theories, and a new
factorization system for finitedimensional algebras over a field that
generalizes (separable, purelyinseparable) factorization for field
extensions.
Regards to all  Max Kelly.
ps:
For those without electronic access, a limited number of printed copies
will shortly be available; please request them soon, so that we can alert
the printer.
Max Kelly.
Date: Thu, 17 Nov 1994 10:18:13 0400 (AST)
From: categories
To: categories
Subject: The categorical theory generated by a limit sketch
Date: Wed, 16 Nov 1994 15:20:51 0500
From: Charles F. Wells
About two weeks ago I announced the availability of a paper
The categorical theory generated by a limit sketch
by Michael Barr and Charles Wells
by gopher from gopher.cwru.edu and by ftp from ftp.cwru.edu. I
have discovered that the copy I posted was not the latest
version. The latest version has now been posted. The
differences are correction of a few minor errors plus some
additional references.

Charles Wells, Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland OH 441067058, USA
Phone 216 368 2880 or 216 774 1926
FAX 216 368 5163
Date: Thu, 17 Nov 1994 17:07:27 0400 (AST)
From: categories
To: categories
Subject: processes and irredundant trees
Date: Thu, 17 Nov 1994 16:28:43 +0000 (GMT)
From: Dusko Pavlovic
The promised companion to the paper I announced last week is now
available. The first version, which I gave to some people, actually
CONTAINED AN ERROR  so please download this version if you have an
old copy.
The file is CCPS.ps, and can be downloaded either from
triples.math.mcgill.ca, or from theory.doc.ic.ac.uk.
Regards,
 Dusko
CONVENIENT CATEGORIES OF PROCESSES AND SIMULATIONS
by Dusko Pavlovic
Abstract.
We show that irredundant trees, used by Dana Scott in the early
sixties, can be used as canonical representants of the bisimilarity
classes of automata (or of transition systems). Simulations then boil
down to tree morphisms. Along the same lines, the categories of
processes modulo the observational and the branching congruences, with
the suitable simulations as morphisms again, are shown to be
isomorphic with certain subcategories of the category of irredundant
trees.
Date: Mon, 12 Dec 1994 20:20:30 0400 (AST)
From: categories
To: categories
Subject: ftp announcement
Date: Mon, 12 Dec 94 14:25:36 EST
From: Michael Makkai
Some weeks ago, I announced two papers, "Generalized sketches as a
framework for completeness theorems", and "Avoiding the axiom of
choice in general category theory" as being available as hard copies.
I now have run out of the copies, which were rather expensive because
of the lengths of the papers. Now, the same papers have been placed on
anonymous ftp at triples.math.mcgill.ca in the directory /pub/makkai
; the first paper is in directory /sketch , the second in /anafun ,
each paper is in several files. They are also available via WWW
ftp://triples.math.mcgill.ca/pub/makkai/makkai_triples.html .
If you need help, there is a README file which you can obtain as
follows:
ftp triples.math.mcgill.ca
cd pub/makkai
get README
bye
Michael Makkai
Date: Wed, 25 Jan 1995 10:07:29 0400 (AST)
Subject: Preprint Available
Date: Tue, 24 Jan 95 17:53:08 +1100
From: Walter Tholen
The following paper may be requested from Maria Manuel Clementino
(paper or electronic copy) by writing to
clementino@gemini.ci.uc.pt
"Compact objects and perfect morphisms"
by M.M. Clementino, E. Giuli and W. Tholen
Abstract:
In a category with a subobject structure and a closure operator,
we provide a categorical theory of compactness and perfectness
which yields a number of classical results of general topology
as special cases, including the product theorems by Tychonoff
and Frolik and the existence of StoneCech compactifications,
both for spaces and maps. Applications to other categories are
also provided.
Date: Mon, 30 Jan 1995 00:52:51 0400 (AST)
Subject: Announce: revision of paper
Date: Fri, 27 Jan 95 11:28:56 EST
From: Robert A. G. Seely
We wish to announce the availability (by ftp or WWW) of the following paper
(revised version)
Natural deduction and coherence
for weakly distributive categories
by
R.F. Blute, J.R.B. Cockett, R.A.G. Seely, and T.H. Trimble
ABSTRACT:
This paper examines coherence for certain monoidal categories using
techniques coming from the proof theory of linear logic, in particular
making heavy use of the graphical techniques of proof nets. We define a two
sided notion of proof net, suitable for categories like weakly distributive
categories which have the twotensor structure (times/par) of linear logic,
but lack a negation operator. Representing morphisms in weakly distributive
categories as such nets, we derive a coherence theorem for such categories.
As part of this process, we develop a theory of expansionreduction systems
with equalities and a term calculus for proof nets, each of which is of
independent interest. In the symmetric case the expansion reduction system
on the term calculus yields a decision procedure for the equality of maps
for free weakly distributive categories.
The main results of this paper are these. First we have proved coherence
for the full theory of weakly distributive categories, extending similar
results for monoidal categories to include the treatment of the tensor
units. Second, we extend these coherence results to the full theory of
*autonomous categories  providing a decision procedure for the maps of
free symmetric *autonomous categories. Third, we derive a conservative
extension result for the passage from weakly distributive categories to
*autonomous categories. We show strong categorical conservativity, in the
sense that the unit of the adjunction between weakly distributive and
*autonomous categories is fully faithful.
NOTES:
This is a significant revision of an earlier version announced a year ago.
The paper has been completely rewritten, and has been enhanced in several
ways:
1) We now treat the cases with nonlogical axioms, and with noncommutative
tensor and par, in considerable detail, pointing out in passing where the
traditional treatment (of the pure commutative case) does not work in these
cases. For example, the traditional proof of sequentiality (via splitting
links) does not work in the presence of nonlogical axioms.
2) We develop a term calculus for proof nets, and a theory of expansion 
reduction rewrite systems, both of which are of independant interest.
3) Using this new material has made the proofs clearer and more complete.
TO OBTAIN THE PAPER:
By ftp: ftp triples.math.mcgill.ca
login as anonymous
use your email address as password
cd pub/rags/nets
binary
get nets.ps.gz (or nets.dvi.gz if you prefer)
(You will have to gunzip this file to get the PostScript  or DVI  file for
the paper.)
By WWW (Mosaic, netscape, ...) My home page is:
ftp://triples.math.mcgill.ca/pub/rags/ragstriples.html
click on the item labelled with the paper's title
 this will display the PostScript file.
(other papers dealing with weakly distributive categories and proof nets
may be found easily via my home page.)
(If you need to get gunzip, you can find it at pip.shsu.edu: issue the
command "quote site index gzip" once you have ftp'd the site to get a
list of files suitable for various computer platforms.)
= rags =
Date: Tue, 7 Feb 1995 01:59:00 0400 (AST)
Subject: Higherdimensional algebra and TQFT
Date: Sun, 5 Feb 95 19:53:07 PST
From: john baez
The paper "Higherdimensional algebra and topological
quantum field theory", by John Baez and James Dolan,
is now available by anonymous ftp as the file
baez/tqft.tex
from
math.ucr.edu
It is in LaTeX, but to LaTeX it you also need the files
auxdefs.sty and diagram.sty, which are also in the directory
baez.
Here's an abstract:
The study of topological quantum field theories increasingly
relies upon concepts from higherdimensional algebra such as
ncategories and nvector spaces. We review progress
towards a definition of ncategory suited for this purpose, and
outline a program in which ndimensional TQFTs are to be
described as ncategory representations. First we describe a
`suspension' operation on ncategories, and hypothesize that
the kfold suspension of a weak ncategory stabilizes for k
greater than or equal to n + 2. We give evidence for this
hypothesis and describe its relation to stable homotopy theory.
We then propose a description of ndimensional unitary extended
TQFTs as weak nfunctors from the `free stable weak ncategory
with duals on one object' to the ncategory of `nHilbert spaces'.
We conclude by describing ncategorical generalizations of
deformation quantization and the quantum double construction.
Date: Wed, 22 Feb 1995 03:56:25 0400 (AST)
Subject: course on categories available
Date: Tue, 21 Feb 1995 15:46:42 +0100
From: Jaap van Oosten
I wrote a first course in category theory which I think more
or less contains what's presumed knowledge in not too specialized
papers and thesises (in computer science). It's 75 pages.
The synopsis is:
1. Categories and functors. Definitions and examples. Duality principle.
2. Natural transformations. Exponents in Cat. Yoneda lemma. Equivalence
of categories; Set^{op} equivalent to Complete Atomic Boolean Algebras.
3. Limits and Colimits. Functors preserving (reflecting) them. (Finitely)
complete categories. Limits by products and equalizers.
4. A little piece of categorical logic. Regular categories, regular epimono
factorization, subobjects. Interpretation of coherent logic in regular
categories. Expressing categorical facts in the logic. Example of
\Omega valued sets for a frame \Omega.
5. Adjunctions. Examples. (Co)limits as adjoints. Adjoints preserve (co)limits. Adjoint functor theorem.
6. Monads and Algebras. Examples. Eilenberg Moore and Kleisli as terminal
and initial adjunctions inducing a monad. Groups monadic over Set.
Lift and Powerset monads and their algebras. Forgetful functor from
TAlg creates limits.
7. Cartesian closed categories and the \lambdacalculus. Examples of ccc's.
Parameter theorem. Typed \lambda calculus and its interpretation in
ccc's. Ccc's with natural numbers object: all primitive recursive functions
are representable
The notes are available by anonymous ftp via:
ftp ftp.daimi.aau.dk
cd pub/BRICS/LS/95/1
get BRICSLS951.ps.gz
Jaap van Oosten
Date: Tue, 21 Mar 1995 04:07:59 0400 (AST)
From: categories
To: categories
Subject: Linear Lauchli Semantics: paper available
Date: Mon, 20 Mar 95 23:42:24 EST
From: SCPSG@acadvm1.uottawa.ca
The paper below is available by anonymous ftp from the
following sites:
triples.math.mcgill.ca, in the directory: pub/blute,
theory.doc.ic.ac.uk, in the directory: papers/Scott.
ftp.csi.uottawa.ca , in the directory: pub/papers/PhilScott
The file is called: lauchli.ps.Z.
Any comments would be greatly appreciated.
Cheers,
Philip Scott
P. S. Of course, you may also contact either of the authors for a
hard copy:
R. F. Blute & P. J. Scott
Dept. of Mathematics
University of Ottawa
585 King Edward
Ottawa, Ont. K1N 6N5
Canada

LINEAR LAUCHLI SEMANTICS
R. F. Blute P. J. Scott
We introduce a linear analogue of Lauchli's semantics for
intuitionistic logic. In fact, our result is a strengthening
of Lauchli's work to the level of proofs, rather than
provability. This is obtained by considering continuous actions
of the additive group of integers on a category of topological
vector spaces. The semantics, based on functorial
polymorphism, consists of dinatural transformations which are
equivariant with respect to all such actions. Such dinatural
transformations are called uniform. To any sequent in
Multiplicative Linear Logic (MLL), we associate a vector space
of ``diadditive'' uniform transformations. We then show that this
space is generated by denotations of cutfree proofs of
the sequent in the theory MLL+MIX. Thus we obtain a full
completeness theorem in the sense of Abramsky and Jagadeesan,
although our result differs from theirs in the use of
dinatural transformations.
As corollaries, we show that these dinatural transformations compose,
and obtain a conservativity result: diadditive dinatural transformations
which are uniform with respect to actions of the additive group of
integers are also uniform with respect to the actions of arbitrary
cocommutative Hopf algebras. Finally, we discuss several possible
extensions of this work to noncommutative logic.
It is well known that the intuitionistic version of
Lauchli's semantics is a special case of the theory of logical
relations, due to Plotkin and Statman.
Thus, our work can also be viewed as a first step towards
developing a theory of logical relations for linear
logic and concurrency.
Date: Mon, 27 Mar 1995 20:31:16 0400 (AST)
From: categories
To: categories
Subject: the thesis, Complexity Doctrines, by web or ftp
Date: Mon, 27 Mar 1995 17:14:24 0500
From: Jim Otto
Dear People,
The thesis, Complexity Doctrines (14+121
pages), contains the chapters
 Tensor and Linear Time
 VComprehensions and P Space
 Dependent Products and Church Numerals
 3Comprehensions and Kalmar Elementary
and was submitted 32895.
It is available by web from either of
ftp://triples.math.mcgill.ca/ctrc.html
ftp://triples.math.mcgill.ca/pub/otto/otto.html
or by ftp from
triples.math.mcgill.ca
/pub/otto/thesis.ps.gz
(E.g. use gunzip and ghostview.)
Bon Soir, Jim Otto
Date: Wed, 10 May 1995 23:03:36 0300 (ADT)
Subject: announcement of paper
Date: Wed, 10 May 1995 20:24:52 +1000 (EST)
From: C. Barry Jay
A Semantics for Shape
=====================
by
C. Barry Jay
is now available by anonymous ftp at
ftp.socs.uts.edu.au
in the file
users/cbj/shape_semantics.ps.Z
Abstract

Shapely types separate data, represented by lists, from shape, or
structure. This separation supports shape polymorphism, where
operations are defined for arbitrary shapes, and shapely operations,
for which the shape of the result is determined by that of the input,
permitting static shape checking. The shapely types are closed under
the formation of fixpoints, and hence include the usual algebraic
types of lists, trees, etc. They also include other standard data
structures such as arrays, graphs and records.
Date: Wed, 17 May 1995 00:58:55 0300 (ADT)
Subject: Announcement of papers
Date: Wed, 17 May 95 10:40:41 +1000
From: Sjoerd Crans
Dear people,
The following papers are now available by anonymous ftp from
ftp.maths.usyd.edu.au, in the directory sydcat/papers/crans.
Hardcopies are also available, requests for this can be sent
to the address below.
thcms.ps
Sjoerd E. Crans
Quillen closed model structures for sheaves,
to appear in {\em Journal Pure Appl. Algebra} 100 (1995).
Abstract:
In this paper I give a general procedure of transferring closed model
structures along adjoint functor pairs. As applications I derive from a
global closed model structure on the category of simplicial sheaves
closed model structures on the category of sheaves of 2groupoids, the
category of bisimplicial sheaves and the category of simplicial sheaves
of groupoids. Subsequently, the homotopy theories of these categories
are related to the homotopy theory of simplicial sheaves.
thpp.ps
Sjoerd E. Crans
Pasting presentations for $\omega$categories.
Abstract:
The pasting theorem showed that pasting schemes are useful in studying
free $\omega$categories. It was thought that their inflexibility with
respect to composition and identities prohibited wider use. This is not
the case: there is a way of dealing with identities which makes it
possible to describe $\omega$categories in terms of generating pasting
schemes and relations between generated pastings, \ie{}, with pasting
presentations. In this paper I develop the necessary machinery for
this. The main result, that the $\omega$category generated by a
pasting presentation is universal with respect to respectable families
of realizations, is a generalization of the pasting theorem.
thten.ps
Sjoerd E. Crans
Pasting schemes for the monoidal biclosed structure on
$\omega\mbox{} \bf Cat$.
Abstract:
Using the theory of pasting presentations, developed in the previous
paper, I give a detailed description of the tensor product on
$\omega$categories, which extends Gray's tensor product on
$2$categories and which is closely related to BrownHiggins's tensor
product on $\omega$groupoids.
Immediate consequences are a general and uniform definition of higher
dimensional lax natural transformations, and a nice and transparent
description of the corresponding internal homs. Further consequences
will be in the development of a theory for weak $n$categories, since
both tensor products and lax structures are crucial in this.
Sjoerd Crans
School of Mathematics and Statistics
University of Sydney
NSW 2006
Australia
email: crans_s@maths.usyd.edu.au
Date: Mon, 26 Jun 1995 09:57:17 0300 (ADT)
Subject: defended and corrected thesis: `Complexity Doctrines'
Date: Sun, 25 Jun 95 21:25:02 EDT
From: James Otto
Dear people,
The thesis `Complexity Doctrines', which was announced to this list at
submission, was defended June 9. The June 13 corrected version is now
linked to
ftp://triples.math.mcgill.ca/pub/otto/otto.html
which is in turn linked to
ftp://triples.math.mcgill.ca/ctrc.html
(The thesis is actually at
ftp://triples.math.mcgill.ca/pub/otto/thesis.ps.gz.)
Bon Soir, Jim Otto
otto@triples.math.mcgill.ca
Date: Tue, 22 Aug 1995 08:38:24 0300 (ADT)
Subject: update to (and correction of) Complexity Doctrines
Date: Sun, 20 Aug 95 20:39:50 EDT
From: James Otto
Dear People,
The following update to (and correction of) my thesis, Complexity
Doctrines, is also linked to (either of)
ftp://triples.math.mcgill.ca/ctrc.html
ftp://triples.math.mcgill.ca/pub/otto/otto.html
Regards, Jim Otto
Update to Complexity Doctrines
J. Otto
August 20, 1995
In this note we
1. point out an (annoying) error (of ours) in background material on
G"odel's system T, and thus pose an question,
2. improve the definition of tier 0,
3. improve the definition (and name) of sketches theories,
4. reduce presheaves to (Makkai) sketches, and
5. propose a definition of resolutions.
We thus update the June 13, 1995 version of Complexity Doctrines.
That version is currently linked to
ftp://triples.math.mcgill.ca/pub/otto/otto.html
By the way, Springer LNCS 953 contains a slightly earlier (with an X
which clearly should be a 1) and abridged version of Chapter 2 of
Complexity Doctrines.
1. System T
We consider NNO (= natural numbers objects) in SMC (=symmetric
monoidal closed) and CC (= cartesian closed) categories. Write (as
this is not TeX) * for tensor and I for the unit. Define o by _ * X
 X o _. Write S for the category of SMC categories having NNO and
of functors preserving chosen structure, and T for the category of CC
categories having NNO and of functors preserving chosen structure.
With J and K initial categories in S and T and with standard structure
on set (the category of small sets), we have the (unique) S and T
functors j : J > set and k : K > set.
Statement 5 of Proposition 1.2.4.1 claims (which we now doubt) that j
and k represent the same numeric functions (= functions between finite
powers of N). The purported proof of Statement 5 fails (as we finally
saw after kind remarks by P. Scott at the 1995 Category Theory and
Computer Science meeting) as while the diagonal diag : N > N * N is
definable in J [Par'e Rom'an], it is doubtful that e.g. the diagonal
diag : N o N > (N o) * (N o N) is definable in J.
As both j and k represent the primitive recursive numeric functions,
they both represent Turing machines modulo how long the machines run.
Thus which numeric functions j and k represent is a matter of how fast
such functions can grow. It is known [Rose, Subrecursion] that the
(set of) numeric functions represented by k is the extended
Grzegorczyk hierarchy below epsilon_0. At least a naive attempt to
obtain such growth of numeric functions represented by j fails. In J
we can define, by commuting diagrams,
0 * N s * N
I * N > N * N > N * N
  
 l_N  +  +
v v v
N > N > N
id s
e_2 = + diag
0 s
I > N > N
  
 id  e_n+1  e_n+1
v v v
I > N > N
s 0 e_n
Now we would like to diagonalize the e_n to get beyond primitive
recursion. But this may be undefinable in J. However in K, by using
diag : N o N > (N o N) * (N o N) and now writing _ x X  X => _
and 1 rather than _ * X  X o _ and I, we can define
0 x (N => N) s x (N => N)
1 x (N => N) > N x (N => N) > N x (N => N)
  
 p_1  p_1, '  p_1, '
v v v
N => N > (N => N) x N > (N => N) x N
id, s 0 ! p_0, @
0 s
1 > N > N
  
 id  f  f
v v v
1 > N => N > N => N
f_2 "
Here _ x X  X => _ defines @ : (X => Y) x X > Y terminal in the
comma category (_ x X)/Y and
e_2 : N > N

f_2 : 1 > N => N
' : N x (N => N) > N

" : (N => N) > (N => N)
e : N x N > N

f : N > N => N
Further ! is the unique map to 1. Then we can diagonalize: e diag.
Thus we pose the question
Which numeric functions are represented by j?
2. Tier 0
Tier 0 can be defined as the pseudoequalizer (aka isoinserter) of
T
C > C
>
I !
(with I ! taking objects to the unit I and maps to the identity on I).
3. Sketches Theories
Sketches theories, formerly sketch theories, are theories of (Makkai
rather than Ehresmann) sketches.
Definition. With a a cardinal, an asketches theory is a category S
such that
1. S is small,
2. S is wellfounded, and
3. S has fanout < a.
Here that S is wellfounded is that for all S objects X all chains of
composable nonidentity maps starting from X
X > > ...
have finite length. Note that wellfounded implies acyclic (aka
1way) and skeletal. Further, S having fanout < a is that for all S
objects X the cardinality of the set (indeed, cone) of maps starting
from X is < a. We are mainly interested in the finitary case, which
is when a is countable.
4. Reducing Presheaves to (Makkai) Sketches.
[Ad'amek Rosick'y] elegantly present the accessible categories, in
various classes, as the full subcategories in categories of presheaves
set^C of the objects [cone] (orthogonal  injective) relative to small
sets A of (cones  maps). We reduce, by orthogonality, the categories
of presheaves set^C to categories of asketches set^S.
Proposition. Given a small category C there is a 4sketches theory S
and a (small) set A of finitely presentable set^S maps such that set^C
is equivalent to the full subcategory in set^S of objects orthogonal
to A.
Proof. As S objects, take the objects and maps of C. As nonidentity
S maps, indexed by the C maps f : X > Y, take the (formal) maps c_f
: f > Y, d_f : f > X. Define a functor G : S > C by c_f > f,
d_f > id_X. Then G^* : set^C > set^S by F > F G is faithful
full. We recover the image of G^*, up to equivalence, by defining A
through the following 3 schemes (using notation as in Complexity
Doctrines). For C maps f : X > Y
{! i_f x : f d_f i_f x = x : X [x : X]}
For C identity maps j : X > X
{c_j a = d_j a : X [a : j]}
For C compositions
h
>
X > Y > Z
f g
{c_h c = c_g b : Z
[d_h c = d_f a : X c_f a = d_g b : Y a : f b: g c : h]}
QED
5. Resolutions
Consider a finitary sketches theory S and a (small) set A of finitely
presentable maps in set^S. Again (as in [Ad'amek Rosick'y]) reduce
orthogonality to injectivity. (Thus, for nonepi maps a in A, add to
A the a^* : P > X induced by the pushout of a along a:
a a
> >
   
 a   a  id
v po v v v
> P > X
id
.) Roughly following [Makkai], define deductions d as compositions of
pushouts (in set^S) of maps a (thought of as axioms) in A.
We propose defining resolutions as cospans (in set^S)


v
>
d
with the left map a deduction. Then resolutions compose as cospans:


d' v
>
 
 
v po v
> >
d
As the initial sketch orthogonal to the (original) A is the colimit of
the diagram of deductions from the empty sketch 0, one may wish to
resolve to 0:


v
0 >
d
In a resolution
X

 f
v
>
d
think of X as a query and of f as a partial answer.
End of note.
Date: Sat, 30 Sep 1995 14:25:17 0300 (ADT)
Subject: Braided monoidal 2categories
Date: Fri, 29 Sep 1995 16:27:23 0700
From: john baez
Martin Neuchl and I have written a paper entitled "Higherdimensional
algebra I: braided monoidal 2categories". In it, we begin with
a brief sketch of what is known and conjectured concerning braided
monoidal 2categories and their relevance to 4d topological quantum
field theories and 2tangles. Then we give concise definitions of
semistrict monoidal and braided monoidal 2categories, and show how
these may be unpacked to give definitions similar to, but not quite
the same as, those given by Kapranov and Voevodsky. Finally, we
describe how to construct a semistrict braided monoidal 2category
as the "center" of a semistrict monoidal 2category, in a manner
analogous to the construction of a braided monoidal category as
the center (or "quantum double") of a monoidal category. As a corollary
this yields a strictification theorem for braided monoidal 2categories.
This paper is available by anonymous ftp to math.ucr.edu. It is
the file bm2cat in the directory baez. It is in uuencoded, compressed
form, because it is 51 pages with lots of pictures. To print it,
first save the file as "bm2cat" and do "uudecode bm2cat". This
should produce a file "bm2cat.dvi.Z". Then do "uncompress bm2cat.dvi.Z".
This should produce a file "bm2cat.dvi," which you can print out
in the way you usually print dvi files.
John Baez
Date: Mon, 30 Oct 1995 16:37:23 0400 (AST)
Subject: Shape papers
Date: Mon, 30 Oct 1995 13:47:03 GMT
From: Barry Jay
The following papers on
SHAPE
may be of interest to category theorists and computer scientists.
The titles are:
"A semantics for shape" shape_semantics.ps.Z
"Data categories and functors" datacats.dvi.Z
"Covariant types" covtypes.dvi.Z
"Polynomial Polymorphism" (in directory P2)
"Typefree term reduction for covariant types" typefree.dvi.Z
"Shape analysis for parallel computing" parshape.dvi.Z
All are available by anonymous ftp from
ftp.socs.uts.edu.au
in the directory
Users/cbj
*Some* can be acessed from my www home page at
http://linus.socs.uts.edu.au/~cbj
The rest of this message describes the main results of the papers, and
some of the goals of the Shape project.
Barry Jay
University of Technology, Sydney
cbj@socs.uts.edu.au
ReplyTo: cbj21@newton.cam.ac.uk
(until 30/11/95)
A semantics for shape
=====================
The basic observation behind shape theory is that most of the functors
F used to model data types share a common characteristic; they have a
cartesian natural transformation into a functor used to store
unstructured data. In the simplest case, the latter is the list (or
free monoid) functor:
data: F => L
The main result of this paper is that in a locos (an extensive
category with all finite limits and lists) all functors shapely over
lists are closed under the formation of initial algebras. The proof is
constructive  simply build a parser for the initial algebra,
using the existing lists and pullbacks.
Data categories and functors
============================
The functors which are cartesian over lists are good for handling
first order structures, but they are not closed under exponentiation,
and so are inadequate for higherorder types. This defect is remedied
by changing the functor used to store data from lists to a *position
functor* given by an object of positions P. Such a functor maps an
object A to the object
P > A+1 .
A *data functor* is a functor F with a given cartesian transformation
to such a position functor. Now, for any object X the functor which
maps A to the object
X > FA
is also a position functor, with object of positions XxP.
The key result about data functors is that every natural
transformation between two such is given by a uniform, or parametric
natural transformation. More precisely, if F is a data functor with
object of positions P, and G is a data functor, then every natural
transformation
F ==> G
is determined by a morphism
F1 > GP
This fact makes the data functors suitable for modelling higher types.
Covariant Types
===============
The data categories, in which the theory of data functors is
developed, include the usual semantic categories, such as Sets, and
bottomless c.p.o.'s. However, Reynolds proved that "Polymorphism is
not settheoretic" by showing that the secondorder polymorphic lambda
calculus (system F) has no settheoretic models. This leads us to ask
what kind of polymorphism is modelled by the data functors. This leads
to the covariant type system in which function types are replaced by
*transformation types*. The system is strong enough to capture the
usual polymorphism of lists and trees, while still having
settheoretic models. Thus,
Polymorphism *can* be settheoretic
Polynomial Polymorphism
=======================
As a subsystem of F, the covariant types do not capture
functoriality. For shape (or functorial) polymorphism to make sense,
there must be a polymorphic algorithm for evaluating the action of
functors on morphisms, i.e. a polymorphic map. Such an algorithm was
first developed in the type system P2, as described in the following
paper.
Typefree term reduction for covariant types
============================================
A generic algorithm for mapping requires the detection of the data to
which the mapped function must be applied. One method of doing this is
to *tag* the data using a single system of tags appropriate for all
the functors under discussion. A naive approach leads to the tagged
types of this paper.
Functorial types
================
Current work aims to have functors represented directly by types so
that, for example, composition of functors is a primitive operation on
types. This is intended to extend the notion of category theory as a
programming technique.
Shape analysis for parallel computing
=====================================
While shape polymorphism allows us to "ignore" the shape, shape
analysis uses detailed shape information to improve errr detection and
compilation. This is particularly important in parallel programming,
where the shape of the data structures, and their distribution, are
central concerns. This paper presents a survey of the issues, and a
computational paradigm, that will be developed by the
Algorithms and Languages Group
University of Technology, Sydney
http://linus.socs.uts.edu.au/~shape
Date: Tue, 7 Nov 1995 21:54:02 0400 (AST)
Subject: announcement
Date: Tue, 7 Nov 1995 20:30:48 0500
From: Michael Makkai
This is to announce a research monograph,
First Order Logic with Dependent Sorts,
with Application to Category Theory
by M. Makkai (McGill Univ.)
(Preliminary version)
Abstract
J. Cartmell [2] introduced a syntax of variable types, which I call
dependent sorts, for the purposes of presenting generalized
algebraic theories. Cartmell's syntax was "abstracted from ...
MartinLof type theory". I add propositional connectives and
quantification to a simplified version of Cartmell's syntax, to
obtain what I call FirstOrder Logic with Dependent Sorts (FOLDS).
The simplification consists in the exclusion of operation symbols,
and a severe restriction on the use of equality. Quantification is
subject to the natural restriction that a quantifier "for all x "
or "there is x " cannot be applied if in the resulting formula
there is a free variable whose sort depends on x .
An important special case of FOLDS was introduced by G. Blanc [1]
for the purpose of characterizing firstorder formulas in the
language of categories that are invariant under equivalence of
categories. P. Freyd's earlier characterization [3], although not
explicitly coached in an instance of FOLDS, is essentially the same
as Blanc's. A. Preller [7] gives an explicit comparison of Blanc's
and Freyd's contexts. The main aim of the present work is to extend
Blanc's and Freyd's characterization from statements about
categories to statements about more complex categorical structures.
A similarity type for structures for FOLDS is given by a oneway
category of sortforming symbols and relation symbols. Oneway
categories were isolated by F. W. Lawvere [4], and were
subsequently shown by him to be relevant for the generalized
sketchsyntax of [5].
The basic metatheory of FOLDS is a simple extension of that of
ordinary multisorted firstorder logic. There are simply formulated
complete formal systems for both classical and intuitionistic
FOLDS, with Kripkestyle completeness for the intuitionistic case.
The systems use entailmentsincontexts as their basic units;
contexts are systems of typings of variables as usual in Martin
Lofstyle systems. We have Gentzenstyle systems admitting cut
elimination. Natural forms of Craig Interpolation and Beth
Definability are true in both classical and intuitionistic FOLDS.
Much of the basic metatheory is done through the formalism of
appropriate fibrations (hyperdoctrines).
The main new concept is a notion of equivalence of structures for
FOLDS. Equivalent structures satisfy the same sentences of FOLDS.
The main general result is that conversely, first order properties
invariant under equivalence are expressible in FOLDS. A stronger
version of the result takes the form of an interpolation theorem.
Two categories are equivalent in the usual sense iff they are
equivalent as structures for FOLDS. This connection between the
categorical concept of equivalence and FOLDSequivalence persists
for more complex categorical structures such as (1) a diagram of
categories, functors and natural transformations, or (2) a
bicategory, or (3) a diagram of bicategories, etc., if we pass to
"ana"versions of the concepts of functor, bicategory, etc.; the
latter were introduced in [6]. Every functor, bicategory, etc., has
its socalled saturation, a simply defined saturated anafunctor,
saturated anabicategory, etc., respectively. A property written in
FOLDS of the saturation is a particular, "good", kind of first
order property of the original.
Applications of the foregoing give syntactical characterizations of
properties invariant under equivalence in the contexts mentioned.
E.g., a firstorder property of a variable bicategory is invariant
under biequivalence iff it is expressible in FOLDS as a property of
the saturated anabicategory canonically associated with the given
bicategory.
References
[1] G. Blanc, Equivalence naturelle et formules logiques en
theorie des categories. Archiv math. Logik 19 (1978), 131137.
[2] J. Cartmell, Generalised algebraic theories and contextual
categories. Annals of Pure and Applied Logic 32 (1986), 209243.
[3] P. Freyd, Properties invariant within equivalence types of
categories. In: Algebra, Topology and Category Theories, ed. A.
Heller and M. Tierney, Academic Press, New York, 1976; pp. 5561.
[4] F. W. Lawvere, More on graphic toposes. Cah. de Top. et Geom.
Diff. 32 (1991), 510.
[5] M. Makkai, Generalized sketches as a framework for
completeness theorems. To appear in J. Pure and Applied Algebra.
[6] M. Makkai, Avoiding the axiom of choice in general category
theory. To appear in J. Pure and Applied Algebra.
[7] A. Preller, A language for category theory in which natural
equivalence implies elementary equivalence of models. Zeitschrift
f. math. Logik und Grundlagen d. Math. 31 (1985), 227234.
(End of Abstract)
A manuscript copy of this work was placed on exhibit at the Category
Theory Meeting in Halifax, 1995; my talk was about the same subject. I
promised to send copies to people who signed up for them. I would
appreciate if those who have been waiting for this, and now find this
announcement, would let me know. I will try to contact those on the
list with me who do not respond.
The manuscript has been placed on anonymous ftp at
triples.math.mcgill.ca in directory /pub/makkai/folds in several
files. You may consult the README file in the directory /pub/makkai.
Date: Fri, 3 Nov 1995 16:21:57 0400 (AST)
Subject: Electronic supplement to ctcs
Date: Fri, 3 Nov 1995 15:00:11 0500
From: Michael Barr
The electronic supplement is now in my ftp directory under the
name that is given in ctcs, namely ctcs.elec.supp.??. There are
actually six forms: {ps,dvi}{ ,gz,zip}.
Date: Fri, 10 Nov 1995 14:45:43 0400 (AST)
Subject: Book Announcement
Date: Fri, 10 Nov 1995 11:17:20 0500
From: Walter Tholen
The book
"Categorical Structure of Closure Operators" by D. Dikranjan and W. Tholen
has appeared in the "Mathematics and Its Applications" series of Kluwer
Academic Publishers (Dordrecht, Boston, London 1995; ISBN 0792337727).
Abstract: The book provides a comprehensive categorical theory of closure
operators, with applications to topological and uniform spaces, groups,
Rmodules, fields and topological groups, as well as to partially ordered
sets and graphs. In particular, closure operators are used to give
solutions to the epimorphism and cowellpoweredness problem in many
concrete categories. The material is illustrated with many examples and
exercises, and open problems are formulated in order to stimulate further
research.

Walter Tholen
Department of Mathematics and Statistics
York University, North York, Ont. Canada M3J 1P3
tel. (416) 736 5250
fax. (416) 736 5757
Date: Fri, 10 Nov 1995 16:39:59 0400 (AST)
Subject: Abstracts "Descent Theory", Oberwolfach '95
Date: Fri, 10 Nov 1995 15:20:10 0500
From: Walter Tholen
Anybody who is interested in getting the files for the Abstracts of talks
given at the meeting on "Geometric and Logical Aspects of Descent Theory"
in Oberwolfach (September '95) may access these by contacting my home page
on the WWW (address below) and clicking on the respective item.
Participants will receive a hardcopy of these abstracts automatically,
sent to them by the Institute.

Walter Tholen
Department of Mathematics and Statistics
York University, North York, Ont., Canada M3J 1P3
tel. (416) 736 5250 or 736 2100, ext. 33918
fax. (416) 736 5757
http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html
Date: Tue, 5 Dec 1995 22:41:33 0400 (AST)
Subject: paper on equivariant homotopy
Date: Mon, 4 Dec 1995 18:06:48 GMT
From: Manuel Bullejos
The following paper can be obained from my www page with address
http:\\www.ugr.es\~bullejos
\title{On the equivariant 2type of a $G$space}
\begin{abstract}
A classical theorem of Mac Lane and Whitehead states that
the homotopy type of a topological space with trivial homotopy at
dimensions 3 and greater can be re\con\struct\ed from its $\pi_1$ and
$\pi_2$, and a cohomology class $k_3\in H^3(\pi_1,\pi_2)$. More recently,
Moerdijk and Svensson suggested the possibility of using Bredon cohomology
to extend this result to the equivariant case, that is, for spaces $X$
equipped with an action by a fixed group $G$. In this paper we carry out
this suggestion and prove an analogue of the classical result in the
equivariant case.
\end{abstract}
Date: Thu, 21 Dec 1995 10:14:00 0400 (AST)
Subject: (Fwd) Re: Abstracts "Descent Theory", Oberwolfach '95
Date: Wed, 20 Dec 1995 13:51:34 0500
From: Walter Tholen
Dear Colleagues,
a link has been appended to my WWW home page to obtain the notes of Ross
Street's lectures on Descent Theory at the Oberwolfach Conference in September.These files may accessed also directly; the address is
ftp://ftp.mpce.mq.edu.au/pub/maths/Categories/Oberwolfach/
The files themselves are:
rwrr 1 ross ftpmaths 1329104 Dec 13 16:34 Oberwolfach_1.ps
rwrr 1 ross ftpmaths 707855 Dec 13 16:33 Oberwolfach_1.ps.Z
rwrr 1 ross ftpmaths 1196252 Dec 13 16:34 Oberwolfach_2.ps
rwrr 1 ross ftpmaths 605741 Dec 13 16:34 Oberwolfach_2.ps.Z
rwrr 1 ross ftpmaths 1018897 Dec 13 16:37 Oberwolfach_3.ps
rwrr 1 ross ftpmaths 549743 Dec 13 16:37 Oberwolfach_3.ps.Z
... giving a PostScript file (txt) and a compressed (binary) version
of each.
A good Web browser can get them using the above as a URL.
It may even automatically uncompress and render the PostScript file.
Best wishes for the Holiday Season and a Happy New Year!
Walter.

Walter Tholen
Department of Mathematics and Statistics
York University, North York, Ont., Canada M3J 1P3
tel. (416) 736 5250 or 736 2100, ext. 33918
fax. (416) 736 5757
http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html
Date: Tue, 23 Jan 1996 21:08:18 0400 (AST)
Subject: draft paper: From Horn formula to Makkai sketch resolution
Date: Mon, 22 Jan 1996 23:42:51 0500
From: James Otto
Dear People,
The plain text draft paper, whose title and abstract follows, is
availabe by web or ftp at
ftp://triples.math.mcgill.ca/pub/otto/res
and is linked to
ftp://triples.math.mcgill.ca/pub/otto/otto.html
Best regards, Jim
From Horn formula to Makkai sketch resolution
J. Otto
January 22, 1996
otto@triples.math.mcgill.ca
Abstract. We provide a basis for logic programming into locally
finitely presentable (or l.f.p.) categories. We thus begin to
consider higher order logic programming. Horn formulas, in particular
systems of equations, generalize to finite Makkai (or M) sketches.
Further, models of sets of Horn clauses generalize to, again called
models and forming the l.f.p. categories, Msketches orthogonal to
sets of maps (axioms) between finite Msketches. Resolutions compute
maps (answers) from finite Msketches (queries) to initial models.
Resolutions are cospans and lift to compositions of special
resolutions.
Date: Mon, 29 Jan 1996 17:09:08 0400 (AST)
Subject: correction to `From ... resolution'
Date: Sat, 27 Jan 1996 08:31:41 0500
From: James Otto
Dear People,
A correction was made to the 12396 version of
ftp://triples.math.mcgill.ca/pub/otto/res
Regards, Jim
12796
Date: Wed, 31 Jan 1996 14:12:27 0400 (AST)
Subject: groupoids
Date: Wed, 31 Jan 1996 09:12:09 0800 (PST)
From: Alan Weinstein
Dear Colleagues,
I've just finished a survey article entitled
"Groupoids: unifying internal and external symmetry", which I have
submitted to the Notices of the AMS. It is available as a postscript
file via email (alanw@math.berkeley.edu) or my web page
(http://math.berkeley.edu/~alanw). Comments are welcome, of course.
Alan Weinstein
Date: Fri, 2 Feb 1996 14:55:38 0400 (AST)
Subject: Preprints
Date: Fri, 2 Feb 1996 10:29:02 +0100
From: Marco Grandis
The following two preprints will soon be available. A "hard" copy will be
sent on request.
With best regards,
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
Via Dodecaneso 35
16146 Genova, Italy
(Email: grandis@dima.unige.it)
***
1. M. Grandis, Categorically algebraic foundations for homotopical algebra,
Dip. Mat. Univ. Genova, Preprint 293 (1996).
Abstract. We investigate a structure for an abstract cylinder endofunctor
I which produces a good basis for homotopical algebra. It essentially
consists of the usual operations (faces, degeneracies, connections,
symmetries, composition) together with a transformation I^2 > I^2,
which we call lens collapse after its realisation in the standard
topological case.
This structure, if somewhat heavy, has the interest of being
"categorically algebraic", i.e. based on operations on functors.
Consequently, it can be naturally lifted from a category A to its
categories of diagrams A^S and its slice categories A\X, A/X. Further,
the dual structure, based on a cocylinder (or path) endofunctor P can be
lifted to the category of Avalued sheaves on a site, whenever P
preserves limits, and to the category of internal monoids in A, with
respect to any monoidal structure of A consistent with P.
2. M. Grandis, On the homotopy structure of strongly homotopy associative
differential algebras, Dip. Mat. Univ. Genova, Preprint 294 (1996).
Abstract. We study here the homotopy structure of Shad, the
category of strongly homotopy associative dalgebras (shadalgebras for
short), also called A_infinityalgebras and introduced by Stasheff ([St],
1963) for the study of the singular complex of the loopspace of a pointed
topological space.
Shad extends the category Da of associative differential
(graded) algebras, by allowing for a homotopy relaxation of objects and
morphisms, up to systems of homotopies of arbitrary degree. The
better known category Dash of associative differential algebras and
strongly homotopy multiplicative maps (StasheffHalperin [StH], Munkholm
[Mu14]), having strict objects (the ones of Da) and lax morphisms (the
ones of Shad) is intermediate between them. A crucial advantage of Shad
over its subcategories Dash and Da is the homotopy invariance property
proved by Gugenheim  Stasheff [GuS].
In order to study shadhomotopies of any order and their
operations, the usual cocylinder functor of dalgebras is here extended to
Shad, where we construct the vertical composition and reversion of
homotopies (also existing in Dash, but not in Da) and homotopy
pullbacks (which exist in Da, but not in Dash).
Shad acquires thus a laxified version of the homotopy structure
studied by the author in previous works; the main results therein,
developing homotopical algebra from the Puppe sequence to stabilisation and
triangulated structures, can very likely be extended to the new axioms, so
to be available for Shad.
Date: Wed, 14 Feb 1996 15:16:30 0400 (AST)
Subject: Preprint (announcement)
Date: Wed, 14 Feb 1996 13:12:33 +0100
From: Anders Kock
The paper "Frame distributions, and support"
by A. Kock and G.E,. Reyes
is available by anonymous ftp :
ftp://theory.doc.ic.ac.uk/papers/Kock
where it appears as distr.ps.Z
It is a slightly expanded version of the preprint with the same title,
Universite de Montreal DMS No 386, janvier 1996.
Abstract: We analyze in terms of constructive frame theory
the relationship between support of distributions,
regularization of opens, and "frame distributions"
(sup lattice maps from a frame to the frame of truth values,
as considered recently by Bunge and Funk).
Date: Thu, 15 Feb 1996 11:13:38 0400 (AST)
Subject: Prolongation by zero (preprint)
Date: Thu, 15 Feb 1996 14:26:36 +0100
From: Anders Kock
The preprint "Locally closed sublocales and prolongation by zero"
is available by anonymous ftp from
theory.doc.ic.ac.uk/papers/Kock
where it appears as the file: locclo.ps (it is about 150 kb).
Abstract: We prove that abelian group valued sheaves over a locally closed
sublocale admit prolongation by zero, and that the prolongation functor has
a right adjoint. The method is by left exact comonads; specifically, it
uses a version of ArtinWraith glueing, where one has to put glue on _both_
the items to be glued.
Date: Tue, 28 Feb 1995 08:43:22 0400 (AST)
Subject: Enrichment and Representation Theorems for Domains.
Date: Mon, 26 Feb 1996 15:06:26 GMT
From: Marcelo Fiore
The prepint "Enrichment and Representation Theorems for Categories of
Domains and Continuous Functions" is available from
http://www.dcs.ed.ac.uk/home/mf files rep.dvi or rep.ps, or by
anonymous ftp from ftp.dcs.ed.ac.uk directory pub/mf files rep.dvi or
rep.ps.

Enrichment and Representation Theorems
for Categories of Domains and Continuous Functions
Synopsis
Domaintheoretic categories are axiomatised by means of categorical
nonordertheoretic requirements on a cartesian closed category
equipped with a commutative monad. We prove an enrichment theorem
showing that every axiomatic domaintheoretic category can be endowed
with an intensional notion of approximation, the path relation, with
respect to which the category Cpoenriches. Subsequently, we provide a
representation theorem of the form: every small domaintheoretic
category (with a lifting monad) has a full and faithful representation
in a domaintheoretic category of cpos and continuous functions (with a
lifting monad) in a suitable intuitionistic set theory.
Our analysis suggests more liberal notions of domains. In particular,
we present a category where the path order is not omegacomplete, but
in which the constructions of domain theory (as, for example, the
existence of uniform fixedpoint operators and the solution of domain
equations) are possible.

Date: Tue, 28 Feb 1995 08:44:04 0400 (AST)
Subject: Paper available by ftp
Date: Tue, 27 Feb 1996 14:17:21 0500
From: Robert A. G. Seely
The following paper is available via ftp or WWW browser, at the URLs
given after the abstract.
Categories for computation in context and unified logic: I
by R.F. Blute, J.R.B. Cockett, and R.A.G. Seely
Abstract
In this paper we introduce the notion of contextual categories. These
provide a categorical semantics for the modelling of computation in context,
based on the idea of separating logical sequents into two zones, one
representing the context over which the computation is occurring, the other
the computation itself. The separation into zones is achieved via a
bifunctor equipped with a tensorial strength. We show that a category with
such a functor can be viewed as having an action on itself. With this
interpretation, we obtain a fibration in which the base category consists of
contexts, and the reindexing functors are used to change the context.
We further observe that this structure also provides a framework for
developing categorical semantics for Girard's Unified Logic, a key feature
of which is to separate logical sequents into two zones, one in which
formulas behave classically and one in which they behave linearly. This
separation is analogous to the context/computation separation above, and is
handled by our semantics in a similar fashion. Furthermore, our approach
allows an analysis of the exponen
ntial structure of linear logic using a
tensorially strong action as the primitive notion. We demonstrate that from
such a structure one can recover a model of the linear storage operator.
Finally, we introduce a sequent calculus for the fragment of Unified Logic
modeled by contextual categories. We show cut elimination for this fragment,
and we introduce a simple notion of proof circuit, which provides a
description of free contextual categories.

Available via browser on my home page
ftp://triples.math.mcgill.ca/pub/rags/ragstriples.html
or directly via ftp
ftp://triples.math.mcgill.ca/pub/rags/bang/context1.[ps,dvi].gz
(The [X,Y] syntax means you should use either X or Y in the URL, not both
and not the "[", "]".)
Date: Mon, 18 Mar 1996 16:32:18 0400 (AST)
Subject: Paper available: Linear Logic complements Classical Logic
Date: Mon, 18 Mar 1996 11:24:50 0800
From: Vaughan Pratt
The paper described below is available by FTP or the web as follows.
FTP
ftp boole.stanford.edu
cd pub
bin
get llcocl.ps.gz
WEB URL:
http://boole.stanford.edu/pub/llcocl.ps.gz
Linear Logic complements Classical Logic
V.R. Pratt
To appear in preliminary proceedings of Linear Logic '96, Tokyo
Classical logic enforces the separation of individuals and predicates,
linear logic draws them together via interaction; these are not
rightorwrong alternatives but dual or complementary logics. Linear
logic is an incomplete realization of this duality. While its
completion is not essential for the development and maintenance of
logic, it is crucial for its application. We outline the
``foursquare'' program for completing the connection, whose corners
are set, function, number, and arithmetic, and define ordinal Set, a
bicomplete *equational* topos, meaning its canonical isomorphisms are
identities, including associativity of product.
This directory also contains 44 other papers on related topics. For a
list of abstracts, see the file ABSTRACTS in this directory, URL:
http://boole.stanford.edu/pub/ABSTRACTS
Vaughan Pratt
Date: Thu, 21 Mar 1996 13:36:59 0400 (AST)
Subject: Money_Games
Date: Thu, 21 Mar 1996 11:48:11 0500 (EST)
From: Andre Joyal
My paper "Free Lattices, communication and money games"
is available on the WWW at:
http://www.math.uqam.ca/_rapports/joyal/Money_Games.html
It is to appear in the Proceedings of the International Congress of
Logic, Methodology and Philosophy of Science
held in Firenze, August 1995.
Andre Joyal
Date: Thu, 21 Mar 1996 23:15:48 0400 (AST)
Subject: Address:Money_Games
Date: Thu, 21 Mar 1996 17:38:11 0500 (EST)
From: Andre Joyal
The correct WWW address for my paper
"Free Lattices, communication and money games"
is:
http://www.math.uqam.ca/_rapports/RapportsTech.html
It is to appear in the Proceedings of the International Congress of
Logic, Methodology and Philosophy of Science
held in Firenze, August 1995.
Andre Joyal
Date: Mon, 25 Mar 1996 10:27:29 0400 (AST)
Subject: more on locally closed sublocales and subtoposes
Date: Mon, 25 Mar 1996 11:26:29 +0100
From: Anders Kock
This is to announce the preprint
A. Kock and T. Plewe: Locally closed subtoposes, and prolongation by zero.
It is available by anonymous ftp from
theory.doc.ic.ac.uk/papers/Kock
where it appears as the file KockPlewe.ps or KockPlewe.dvi .
It subsumes the paper by Kock, announced here on Feb. 15 1996,
and this paper has therefore now been removed from the above ftpsite.
Abstract: We prove the equivalence of some conditions on a complemented
subtopos of a topos, one of which is that the subtopos is locally closed,
and another is that abelian group objects in the smaller topos admit
prolongation by zero; the prolongation functor then has a right adjoint.
Date: Fri, 19 Apr 1996 14:21:32 0300 (ADT)
Subject: My talk at Penn
Date: Fri, 19 Apr 1996 11:41:28 0400
From: Michael Barr
A number of people have written to ask if the paper I am talking
about at Penn is available. Yes. In the ftp directory of
triples, in pub/barr/newasymm.dvi and .ps.
Michael
Date: Tue, 28 May 1996 12:12:19 0300 (ADT)
Subject: revised `From Horn clause to Makkai sketch resolution'
Date: Mon, 27 May 1996 17:14:57 0400
From: James Otto
Dear people,
The May 27, 1996 revision of `From Horn clause to Makkai sketch
resolution' is now at (linked to, indirectly linked to)
ftp://triples.math.mcgill.ca/pub/otto/res
ftp://triples.math.mcgill.ca/pub/otto/otto.html
ftp://triples.math.mcgill.ca/ctrc.html
1. Axiom templates are gone. This simplifies the main result 
lifting (for l.f.p. logic programming)  and eliminates a section.
2. The examples are greatly improved. Even and + are added and head
consolidation is gone.
3. Some terminology is improved and more motivation is added.
Regards, Jim Otto
Date: Tue, 4 Jun 1996 22:23:37 0300 (ADT)
Subject: Preprint (announcement)
Date: Tue, 4 Jun 1996 11:57:15 +0200
From: Anders Kock
The paper "Remarks on the Bianchi Identity"
by A. Kock
is available (compressed dvi or ps versions) from the
Hypatia archive in London
http://hypatia.dcs.qmw.ac.uk
It uses synthetic differential geometry to understand the
Bianchi identity in terms of combinatorial groupoid theory.
Date: Sun, 9 Jun 1996 20:41:04 0300 (ADT)
Subject: May 28, '96 corrected revised `... Makkai sketch resolution'
Date: Sat, 8 Jun 1996 14:45:43 0400
From: James Otto
Dear people,
In the May 27, '96 revised `... Makkai sketch resolution' I implied
that showing that f.p. presheaves over the opposites of signature
categories have projective covers was not besides the point. But, as
an anonymous referee indicated, it seems that it is as it seems that
all f.p. presheaves have projective covers. This was fixed in the May
28, '96 corrected revised `... Makkai sketch resolution' which was
linked then to the URL
ftp://triples.math.mcgill.ca/pub/otto/otto.html
Regards, Jim Otto
Date: Wed, 24 Jul 1996 10:43:31 0300 (ADT)
Subject: paper
Date: Wed, 17 Jul 1996 18:07:50 +0100 (BST)
From: Dusko Pavlovic
Dear Categories,
The following preprint by me (to appear in MSCS) is available via
http://www.cogs.susx.ac.uk/users/duskop/index.html
or by anonymous ftp from
ftp.cogs.susx.ac.uk
file pub/users/duskop/CLNA.ps.gz.
Regards,
 Dusko Pavlovic
CATEGORICAL LOGIC OF NAMES AND ABSTRACTION
IN ACTION CALCULI
Abstract.
Milner's action calculus implements abstraction in monoidal
categories, so that familiar lambdacalculi can be subsumed together
with the picalculus and the Petri nets. Variables are generalised to
*names*: only a restricted form of substitution is allowed.
In the present paper, the wellknown categorical semantics of the
lambdacalculus is generalised to the action calculus. A suitable
functional completeness theorem for symmetric monoidal categories is
proved: we determine the conditions under which the abstraction is
definable. Algebraically, the distinction between the variables and
the names boils down to the distinction between the transcendental and
the algebraic elements. The former lead to polynomial extensions, like
e.g. the ring Z[x], the latter to algebraic extensions like
Z[\sqrt{2}] or Z[i].
Building upon the work of P.~Gardner, we introduce *action
categories*, and show that they are related to the static action
calculus exacly as cartesian closed categories are related to the
lambdacalculus. Natural examples of this structure arise from
allegories and cartesian bicategories. On the other hand, the free
algebras for any commutative Moggi monad form an action category. The
general correspondence of action calculi and Moggi monads will be
worked out in a sequel to this work.
Date: Wed, 24 Jul 1996 14:16:45 0300 (ADT)
Subject: Answer book for CTCS, 2nd Ed.
Date: Wed, 24 Jul 1996 10:57:36 0400
From: Michael Barr
In the second edition, the answer book was omitted (trying, not with a
great deal of success, to hold the price down). We thought it was
going to be published separately, but instead the publisher was simply
sending a photocopy to all who requested it. All who knew to request
it, in fact. So with their permission, we have posted it. On triples,
it will be in usual place (~ftp/pub/barr) as ctcs.ansbook.[dvi,ps].
[ ,zip,gz].
Michael
Date: Wed, 31 Jul 1996 14:56:56 0300 (ADT)
Subject: Answers to "Category Theory for Computing Science"
Date: Tue, 30 Jul 1996 12:22:37 0400
From: Charles Wells
We have made the answers to the second edition of "Category
Theory for Computing Science" available on the web, in two ways.
1) As Mike Barr already announced, you can get them by anonymous FTP, at
triples.math.mcgill.ca in /pub/barr, files ctcs.ansbook.*. These are
available in TeX DVI and Postscript formats.
2) You can also get them by webserver, at
http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/pub/papers.html#ansbook
These are available in DVI, Postscript and Acrobat Reader formats.
Charles Wells, Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland, OH 440167058. Office phone: 216 368 2893.
Math dept phone: 216 368 2880. Fax: 216 368 5163. Home phone: 216 774 1926.
Home Page URL: http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/home.html.
"Some have said that I cannot sing; but no one will say that I didn't sing."
Florence Foster Jenkins
Date: Sun, 18 Aug 1996 11:36:15 0300 (ADT)
Subject: BOOK: Foundations for Programming Languages (Mitchell)
Date: Fri, 16 Aug 1996 11:44:40 0700
From: John C. Mitchell
BOOK ANNOUNCEMENT

Foundations for Programming Languages
by John C. Mitchell
"Programming languages embody the pragmatics of designing
software systems, and also the mathematical concepts which underlie
them. Anyone who wants to know how, for example, objectoriented
programming rests upon a firm foundation in logic should read this
book. It guides one surefootedly through the rich variety of basic
programming concepts developed over the past forty years."
 Robin Milner, Professor of Computer Science, The Computer
Laboratory, Cambridge University
"Programming languages need not be designed in an intellectual
vacuum; John Mitchell's book provides an extensive analysis of the
fundamental notions underlying programming constructs. A basic
grasp of this material is essential for the understanding, comparative
analysis, and design of programming languages."
 Luca Cardelli, Digital Equipment Corporation
Written for advanced undergraduate and beginning graduate students, Foundations for
Programming Languages uses a series of typed lambda calculi to study the axiomatic,
operational, and denotational semantics of sequential programming languages. Later
chapters are devoted to progressively more sophisticated type systems.
Compared to other texts on the subject, Foundations for Programming Languages is
distinguished primarily by its inclusion of material on universal algebra and algebraic
data types, imperative languages and FloydHoare logic, and advanced chapters on
polymorphism and modules, subtyping and objectoriented concepts, and type
inference. The book is mathematically oriented but includes discussion, motivation,
and examples that make the material accessible to students specializing in software
systems, theoretical computer science, or mathematical logic.
Foundations for Programming Languages is suitable as a reference for professionals
concerned with programming languages, software validation or verification, and
programming, including those working with software modules or objectoriented
programming.
MIT Press
Foundations of Computing series
September 1996
ISBN 0262133210
608 pp. << actually 850 pages >>
$60.00 (cloth)
MIT PRESS display: http://wwwmitpress.mit.edu:80/mitp/recentbooks/comp/mitfh.html
Date: Fri, 30 Aug 1996 09:17:54 0300 (ADT)
Subject: 2Hilbert spaces
Date: Thu, 29 Aug 1996 15:31:14 0700 (PDT)
From: john baez
Here is the abstract of a paper I wrote:
HigherDimensional Algebra II: 2Hilbert Spaces
A 2Hilbert space is a category with structures and properties
analogous to those of a Hilbert space. More precisely, we define a
2Hilbert space to be an abelian category enriched over Hilb with a
*structure, conjugatelinear on the homsets, satisfying
= = . We also define monoidal, braided monoidal,
and symmetric monoidal versions of 2Hilbert spaces, which we call
2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and
we describe the relation between these and tangles in 2, 3, and 4
dimensions, respectively. We prove a generalized DoplicherRoberts
theorem stating that every symmetric 2H*algebra is equivalent to the
category Rep(G) of continuous unitary finitedimensional representations
of some compact supergroupoid G. The equivalence is given by a
categorified version of the Gelfand transform; we also construct a
categorified version of the Fourier transform when G is a compact
abelian group. Finally, we characterize Rep(G) by its universal
properties when G is a compact classical group. For example, Rep(U(n))
is the free connected symmetric 2H*algebra on one even object of
dimension n.
This paper is long and contains pictures and diagrams, so I have
made a compressed Postscript file of it available at
http://math.ucr.edu/home/baez/2hilb.ps.Z
It is also available by anonymous ftp to math.ucr.edu, where it is the
file 2hilb.ps.Z in the directory pub/baez. On UNIX systems, at least,
one can download it and then uncompress it by typing
uncompress 2hilb.ps.Z
If any of this presents a problem, email your address to
baez@math.ucr.edu and I can send you hardcopy. I look forward to
comments, criticisms, and corrections.
Date: Mon, 2 Sep 1996 15:55:57 0300 (ADT)
Subject: flexible sheaves
Date: Fri, 30 Aug 1996 10:18:54 0400 (EDT)
From: James Stasheff
If it hasn't been mentionned here already
Carlos Simpson has just posted Flexible sheaves
qalg/9608025
[at http://eprints.math.duke.edu/qalg/  RR]
Jim Stasheff jds@math.unc.edu
MathUNC (919)9629607
Chapel Hill NC FAX:(919)9622568
275993250
http://www.math.unc.edu/Faculty/jds
May 15  August 15:
146 Woodland Dr
Lansdale PA 19446 (215)8226707
Date: Tue, 3 Sep 1996 17:22:30 0300 (ADT)
Subject: Revision of paper on ftp
Date: Tue, 3 Sep 1996 16:11:05 0400
From: Robert A. G. Seely
We wish to announce the following (revised) paper now available on
triples:
CATEGORIES FOR COMPUTATION IN CONTEXT AND UNIFIED LOGIC
by R.F. Blute, J.R.B. Cockett, and R.A.G. Seely
ABSTRACT
In this paper we introduce context categories to provide a framework for
computations in context. The structure also provides a basis for
developing the categorical proof theory of Girard's unified logic. A key
feature of this logic is the separation of sequents into classical and
linear zones. These zones may be modelled categorically as a
context/computation separation given by a fibration. The perspective
leads to an analysis of the exponential structure of linear logic using
strength (or context) as the primitive notion.
Context is represented by the classical zone on the left of the turnstile
in unified logic. To model the classical zone to the right of the
turnstile, it is necessary to introduce a notion of cocontext. This
results in a fibrational fork over context and cocontext and leads to
the notion of a bicontext category. When we add the structure of a
weakly distributive category in a suitably fork fibred manner, we obtain
a model for a core fragment of unified logic.
We describe the sequent calculus for the fragment of unified logic
modelled by context categories; cut elimination holds for this fragment.
Categorical cut elimination also is valid, but a proof of this fact is
deferred to a sequel.
REMARKS
This is a completely revised version of the paper we announced in
February this year. At the suggestion of an anonymous referee, we have
dropped all reference to the circuits of the system we originally
described, and extended the system to include multipleconclusions as
well as multiplehypotheses in the sequent calculus, in both the
"classical" and "linear" positions. This is the heart of the system LU
of Girard, and a recipe is given to allow further extensions. We plan
to describe the circuits (proof nets) for the expanded system in a
sequel, which will allow shorter and clearer proofs of categorical cut
elimination, as well as comparisons with other systems (such as our own
weakly distributive categories with storage and Bierman's MELL).
Since the original paper contains some material not carried over to the
revision  most significantly, the sequent calculus for the single
conclusion logic (the "intuitionist" case), plus the proof circuits for
that logic  it remains on the ftp site with a different name and link.
FTP and WWW locations:
The paper may be found at this URL:
ftp://triples.math.mcgill.ca/pub/rags/bang/context1.[dvi,ps].gz
(The earlier version is ...context0... at the same place.)
You can also get it from my WWW page:
http://www.math.mcgill.ca/~rags
As you can see from the URL above, the dvi and ps files are gzipped 
if you save the files (in binary) format, gunzip them with the command
gunzip  email me if you need help.
Rick Blute
Robin Cockett
Robert Seely
( contact person for ftp help:
rags@math.mcgill.ca )
Date: Wed, 4 Sep 1996 14:42:53 0300 (ADT)
Subject: preprint: Minimal Realization in Bicategories of Automata
Date: Tue, 3 Sep 1996 17:18:20 0300 (ADT)
From: Bob Rosebrugh
This is to announce that the article whose abstract follows is available
at
ftp://sun1.mta.ca/pub/papers/rosebrugh/mnrl.dvi
or from my Web page
http://www.mta.ca/~rrosebru/
Regards to all,
Bob Rosebrugh
==============================================================================
Minimal Realization in Bicategories of Automata
R. Rosebrugh, N. Sabadini and R. F. C. Walters
The context of this article is the program to develop monoidal
bicategories with a feedback operation as an algebra of processes, with
applications to concurrency theory. The objective here is to study
reachability, minimization and minimal realization in these bicategories.
In this setting the automata are 1cells in contrast with previous studies
where they appeared as objects. As a consequence we are able to study the
relation of minimization and minimal realization to serial composition of
automata using (co)lax (co)monads. We are led to define suitable
behaviour categories and prove minimal realization theorems which extend
classical results.
Date: Wed, 4 Sep 1996 14:44:53 0300 (ADT)
Subject: preprints available
Date: Wed, 4 Sep 1996 16:00:06 +0200 (MET DST)
From: koslowj@iti.cs.tubs.de
I've finally given in and created a home page:
http://www.iti.cs.tubs.de/TIINFO/koslowj/koslowski.html
Two recent preprints of interest are:
 A convenient category for games and interaction (15 pages)
(Workshop Domains II, Braunschweig, May 1996 and
PSSL 61, Dunkerque, June 1996)
 Monads and Interpolads in bicategories (29 pages, uses string diagrams)
(CT95, Halifax, July 1995 and in much revised form
Sussex, July 1996)
Other papers will be added in the next few weeks. The abstracts
follow below:
%% Abstract for: A convenient category for games and interaction
We present a simple construction of an orderenriched category
gam that simultaneously dualizes and parallels the familiar
construction of the category rel of relations. Objects of
gam are sets, and arrows are games, viewed as special kinds of
trees. The quest for identities for the composition of trees
naturally leads to the consideration of alternating sequences and
games of a specific polarity. gam may be viewed as a canonical
extension of rel , and just as for rel , the maps in gam
admit a nice charactrization. Disjoint union of sets induces a
special tensor product on gam that allows us to recover the
monoidal closed category of games and strategies of interest in game
theory. If we allow games with explicit delay moves, the
categorical description of the structure that leads to the monoidal
closed category is even more satisfying. In particular, we then
obtain an explicit involution.
%% Abstract for: Monads and interpolads in bicategories
Monads may be viewed as lax functors from the terminal category into
a bicategory. If the target has local stable coequalizers, monads
together with lax functors from the twoelement chain, here called
mmodules, can be organized into another bicategory, through which
every lax functor into the original one factors. Mmodules are
special cases of modules between endo1cells, which behave well
with respect to composition, but in general fail to have identities.
To overcome this problem, we do not need to impose the full
structure of a monad on the endo1cells, an associative
coequalizing operation suffices. The bicategory of these socalled
interpolads together with structurepreserving modules is
Cauchycomplete, and contains the bicategory of monads as a usually
nonfull subbicategory. If we start from a bicategory that has all
right liftings, modules in general, and the bicategories of
interpolads and of monads in particular, inherit this property,
provided the homcategories of the base have equalizers. While
interpolads over rel are just idempotent relations, over the
suspension of set they correspond to interpolative semigroups,
and over spn they lead to a notion of ``category without
identities'', also known as a taxonomy.

J\"urgen Koslowski % Stupidity is the basic building block
ITI % of the universe.
TU Braunschweig %
koslowj@iti.cs.tubs.de % (Frank Zappa)
Date: Thu, 5 Sep 1996 14:11:33 0300 (ADT)
Subject: Correction to paper  distributive is not weakly distributive
Date: Tue, 3 Sep 1996 16:13:35 0400
From: Robert A. G. Seely
The following notice and discussion amplifies some recent remarks made
by Robin Cockett on the CATEGORIES list.
We wish to announce a correction to a statement in the paper
Weakly distributive categories
by J.R.B. Cockett and R.A.G. Seely
An error in Proposition 3.1, where we claimed that distributive
categories are weakly distributive, was found in proof. The result is
totally incorrect: a distributive category is a cartesian weakly
distributive category if and only if it a preorder. (Note: a weakly
distributive category may be cartesian  by which we just mean the
tensor and cotensor ("par") are cartesian product and coproduct
respectively  without being a preorder; it is the distributivity that
causes the collapse.)
In particular, any distributive category which satisfies equation (13):
\delta^R_R
(A+B)x(C+D) > A+(Bx(C+D))
 
\delta^L_L  
v 
((A+B)xC)+D  1 + \delta^L_L
 
\delta^R_R + 1  
v a v
(A+(BxC))+D > A+((BxC)+D)
(where we write x for the tensor, + for the cotensor (par),
and 1 for identity)
for the choice of weak distributions described in the paper is
immediately a preorder. This because in that diagram if A=D=1 and B=C=0
then, up to equivalence, we obtain for the two sides of diagram the
coproduct embeddings of 1 into 1+1. This suffices to cause collapse.
The argument can be modified to show that in any distributive category
which is simultaneously weakly distributive (no matter how the weak
distributions are defined), Boolean negation must have a fixed point.
This also suffices to cause collapse.
A consequence of this observation is that the categorical proof theory
of notnecessarilyintuitionist AND/OR logic is somewhat subtle. In the
absence of any connective for implication, there is no apparent a priori
reason not to have multipleconclusion sequents; let's see what this
yields. We start with the premise that a good semantics for AND/OR
logic ought to be a polycategory; in particular, that the morphisms
interpreting the following two derivations must be equal. (That these
are equal is a consequence of the polycategory definition, but you can
judge them on their own merits if you like. This type of permutation of
cuts is pretty standard, and categorical cut elimination then would
demand that they be equal.)
(Notation: I use > for the sequent turnstile, and x and + for AND and
OR. The interpretation of the commas is, as is usual in such logics,
AND on the left and OR on the right, so there are evident identity maps
representing A,B > AxB and A+B > A,B. All deduction steps are cuts.
The cut rule is XX,A > YY and WW > A,UU entail XX,WW > YY,UU
and variants via exchange.)
B,C > BxC A+B > A,B

A+B,C > A,BxC C+D > C,D

A+B,C+D > A,BxC,D
B,C > BxC C+D > C,D

= B,C+D > BxC,D A+B > A,B

A+B,C+D > A,BxC,D
But here's the catch  with the obvious interpretation, these come out
different in SETS: think of the image of a pair in (A+B)x(C+D),
where a \in A and d \in D. For the top map, this is mapped to a,
whereas for the bottom map it is mapped to d. This is just our equation
(13) again, so the point of our initial comment is that in any
distributive category, with any interpretation, these two maps are equal
iff the category is a preorder.
This is a pretty "stripped down" example  it seems that categorical cut
elimination is inconsistent with using distributive categories for
AND/OR logic and general sequents. This problem is averted of course if
one restricts oneself to "intuitionist" sequents (with the right of the
turnstile restricted to single formulas), but then this result may be
seen as indicating how the folkloric result concerning the collapse of
categorical proof theory for classical logic (Joyal) doesn't really
depend on very much structure  note that we have assumed no structure
rules beyond cut, and the linear versions of the AND/OR sequent rules;
the collapse just needs multipleconclusion sequents and distributivity.
It is interesting to note, however, that by carefully choosing the weak
distributions one can construct a cartesian weakly distributive category
from an elementary distributive category by simply passing to the
Kleisli category of the ``exception monad'' E(X) = X+1. So, for example,
although SETS is not weakly distributive itself, POINTED_SETS is.
The error means, of course, that all discussion in the paper of
nonposetal distributive categories as examples of weakly distributive
categories must be discounted. This mainly affects the Introduction and
Section 3, where Proposition 3.1 must be restated as indicated above,
and the surrounding text must take this restatement into account. In
particular, Theorem 3.3, although still correct, ought to be stengthened
to state that a cartesian weakly distributive category is a preorder if
and only if it has a strict initial object.
A version of this paper which contains a rewritten Introduction and
Section 3 may be found on rags' WWW home page at this URL:
.
These comments will also appear in the published version of the paper
(to appear in JPAA).
Finally, the inevitable controversy about terminology: we have decided
to continue calling these categories "weakly distributive", since we
have done so for so long and in so many places. Besides, Hyland and
dePaiva had arrived at the same name for the "weak distributivities",
independently, and at the same time. But we keep an open mind about
these matters: if another name seems to have nearuniversal approval, we
will adopt it too. The most promising seems to be Barr's suggestion of
"linearly distributive". Indeed, had that suggestion been made in 1991,
we might have adopted it then (it certainly beats "dissociative
categories"!)
Robin Cockett
Robert Seely
(for ftp help: rags@math.mcgill.ca)
Date: Mon, 9 Sep 1996 11:25:43 0300 (ADT)
Subject: Change of address, paper
Date: Sun, 8 Sep 1996 9:20:29 GMT
From: MHEBERT@acs.auc.eun.eg
Here is the abstract of a paper recently accepted in JPAA, available on
request (by mail):
Purity and injectivity in accessible categories
Michel Hebert
Abstract. We introduce a strenghtening of the concept of purity which
may be more appropriate in the context of accessible categories (and
which we prove to be equivalent to the usual one in locally presentable
categories). We then use it to obtain a characterization of (cone)
injectivity classes which, in particular, provides a solution to the
corresponding problem of L.Fuchs(in the context of Abelian Groups) which
avoids the settheoretic assumptions of the AdamekRosicky solution.
Please note that I have also a (slightly modified) new email address:
mhebert@acs.auc.eun.eg
Michel Hebert
Date: Mon, 9 Sep 1996 11:24:46 0300 (ADT)
Subject: Noncommutative Full Completeness: paper available
Date: Sat, 7 Sep 1996 14:02:37 0400
From: Phil Scott
The following paper is available by anonymous ftp from the sites:
triples.math.mcgill.ca in directory pub/blute
theory.doc.ic.ac.uk in directory theory/papers/Scott
Both A4 and North American sized versions are given in gzipped form
as shufA4.ps.gz and shuf.ps.gz. Of course, anyone having problems
can contact either of the authors at the email addresses below.

The Shuffle Hopf Algebra and
Noncommutative Full Completeness
by R.F. Blute and P.J. Scott
email: rblute@mathstat.uottawa.ca, phil@csi.uottawa.ca
ABSTRACT
We present a full completeness theorem for the multiplicative
fragment of a variant of noncommutative linear logic known
as cyclic linear logic (CyLL), first defined by Yetter.
The semantics is obtained by considering dinatural transformations
on a category of topological vector spaces which are equivariant
under certain actions of a noncocommutative Hopf algebra, called the
shuffle algebra. Multiplicative sequents are assigned a vector
space of such dinaturals, and we show that the space has the denotations
of cutfree proofs in CyLL+MIX as a basis. This can be viewed as a fully
faithful representation of a free *autonomous category,
canonically enriched over vector spaces.
This work is a natural extension
of the authors' previous work, ``Linear Lauchli Semantics'', where a
similar theorem is obtained for the commutative logic. In that paper, we
consider dinaturals which are invariant under certain actions of the
additive group of integers. We also present here a simplification of that
work by showing that the invariance criterion is actually a consequence
of dinaturality. The passage from groups to Hopf algebras corresponds
to the passage from commutative to noncommutative logic.
Date: Wed, 18 Sep 1996 11:55:55 0300 (ADT)
Subject: papers available by ftp
Date: Wed, 18 Sep 1996 13:45:24 +0200 (MET DST)
From: Jiri Rosicky
My recent papers are available through anonymous ftp at
ftp.math.muni.cz in the directory /pub/math/people/Rosicky/papers
Jiri Rosicky
Date: Wed, 25 Sep 1996 10:39:29 0300 (ADT)
Subject: Announcement of paper: FILL
Date: Tue, 24 Sep 1996 13:00:00 0400
From: Robert A. G. Seely
We wish to announce the following paper made available for ftp on our
WWW site.
Proof theory for full intuitionistic linear logic,
bilinear logic, and mix categories
by
J.R.B. Cockett and R.A.G. Seely
ABSTRACT
This note is a survey of techniques we have used in studying coherence
for monoidal categories with two tensors, corresponding to the tensor 
par fragment of linear logic. We apply these ideas to several
situations which extend our previous work, in particular, the Full
Intuitionistic Linear Logic (FILL) of Hyland and de Paiva, and the
Bilinear Logic of Lambek. Note that the latter is a noncommutative
logic; we also consider the noncommutative version of FILL. We show
that the essential difference between FILL and multiplicative linear
logic lies in making a tensorial strength natural transformation an
isomorphism. We briefly discuss the structure of the nucleus of a
category modelling FILL: the nucleus is a *autonomous full subcategory.
In addition, we define and study the appropriate categorical structure
corresponding to the mix rule. For all these structures, we do not
restrict consideration to ``pure'' logic, in that we allow for the
inclusion of nonlogical axioms. We define the appropriate notion of
proof nets for these logics, and use them to describe coherence results
for the corresponding categorical structures.
We would draw your attention to the following "highlights":
 we develop proof nets for FILL (as well as bilinear logic  but this
latter is shown equivalent to the system of noncommutative *autonomous
categories we studied in an earlier paper [BCST],
so that is not new).
 we show several equivalent formulations of bilinear logic =
noncommutative $*$autonomous categories. One interesting one is that
if one requires of a FILL category that a natural transformation
equivalent to the tensorial strength given by the weak distributivity is
isomorphic, then you get the full bilinear logic.
 we introduce a generalisation of the notion of "nuclear map" suitable
for weakly distributive categories, and show the nucleus of a FILL
category is *autonomous.
 we give a rigorous definition of what it means for a category to
satisfy the MIX rule (previous attempts dealt only with the existence of
the required maps, and not the necessary coherence also needed), and
prove a coherence theorem for this doctrine.
 these last two points are linked by the observation that a weakly
distributive category satisfies MIX iff its nucleus does. As a
consequence we note that "cartesian" weakly distributive categories
(where the tensor is cartesian product) must satisfy MIX.
The paper is available by ftp at the URL
or
as well as on the home page
For assistance with ftp, please contact rags@math.mcgill.ca.
Robin Cockett
Robert Seely
Date: Thu, 10 Oct 1996 21:59:00 0300 (ADT)
Subject: New paper
Date: Thu, 10 Oct 1996 10:56:10 GMT
From: MHEBERT@acs.auc.eun.eg
Here is the abstract of a paper recently accepted in Annals of Pure and Applied
Logic, available on request (by mail):
Syntactic characterization of closure under pullbacks
and of locally polypresentable categories
Michel Hebert
Abstract: We give syntactic characterizations of (1): the (finitary)
theories whose categories of models are closed under the formation
of pullbacks, and of (2) (its categorical counterpart): the locally
omegapolypresentable categories. A somewhat typical example is the
category of algebraically closed fields. Case (1) is proved by
classical modeltheoretic methods; it solves a problem raised by H.
Volger (with motivations from the theory of abstract data types).
The solution of case (2) is in the spirit of the ones for the locally
omegapresentable and omegamultipresentable cases found by M.
Coste and P.T. Johnstone respectively. The problem (2) was raised in
the context of Domain Theory by F. Lamarche.
This implies in particular the (maybe already known) fact that a
(finitary) theory invariant under pullbacks has its category of
models omegaaccessible. We give an example showing that this is
false for equalizers.
(Note: The paper was written (and submitted) more than 3 years ago)
Michel Hebert
Date: Wed, 16 Oct 1996 11:59:13 0300 (ADT)
Subject: PREPRINTS AVAILABLE
Date: Wed, 16 Oct 1996 11:09:57 +0200
From: I. Moerdijk
I would like to announce that psfiles for the following two recent preprints
are available from
1. I. Moerdijk, Proof of a Conjecture of A. Haefliger.
[Summary: It is proved that for any etale topological groupoid G, and any
abelian Gsheaf A, there is a natural isomorphism H*(G,A) = H*(BG,A') for the
associated sheaf A' on the classifying space.]
2. C. Butz, I. Moerdijk, Topological representation of sheaf cohomology of
sites.
[Summary: We construct for each topos T with enough points a topological space
X(T) and a morphism p from the topos of sheaves on X(T) to T which is acyclic,
i.e. induces a full embedding of derived categories from D(T) into D(X(T)).]

Date: Wed, 16 Oct 1996 12:01:28 0300 (ADT)
Subject: New Paper Available
Date: Wed, 16 Oct 1996 08:34:46 0400
From: Phil Scott
The following paper is available by anonymous ftp from the following
sites:
theory/doc/ic.ac.uk in directory theory/papers/Scott
Homepage of P. Dybjer: http://www.cs.chalmers.se/~peterd
Homepage of P.J.Scott: http://www.csi.uottawa.ca/~phil/extra/papers
=======================================================================
Normalization and the Yoneda Embedding
by
Djordje Cubric, Peter Dybjer, and Philip Scott
We show how to solve the word problem for simply typed
\lambda\beta\etacalculus by using a few wellknown facts about
categories of presheaves and the Yoneda embedding. The formal setting
for these results is $\cP$category theory, a version of ordinary
category theory where each homset is equipped with a partial
equivalence relation. The part of $\cP$category theory we develop
here is constructive and thus permits extraction of programs. It is
this intuitionistic aspect of our work which is fundamental to
obtaining a normalization algorithm.
In a certain sense, our program is dual to J. Lambek's original goal
of categorical proof theory, in which he used cutelimination to study
categorical coherence problems. Here, we use a method inspired from
categorical coherence proofs to normalize lambda terms (and thus
intuitionistic proofs). It is important to stress that in our method, we
make no use of traditional prooftheoretic or rewriting techniques.
Date: Mon, 21 Oct 1996 11:51:34 0300 (ADT)
Subject: preprint
Date: Mon, 21 Oct 1996 12:02:57 +0100
From: Till Plewe
The following preprint (abstract see below) is now available either via ftp at
theory.doc.ic.ac.uk as papers/Plewe/LocQuot.ps.gz or LocQuot.dvi.gz
or at my home page
http://theory.doc.ic.ac.uk/~tp5/index.html

"Quotient maps of locales"
This paper considers the question of how to characterize regular epis in the
category Loc of locales.
The most interesting results are that regular epis don't compose in Loc, and
that not all closed surjections are regular epis, although those with subfit
domain (= all open sublocales are joins of closed sublocales) are.
The paper also contains a characterization of regular epis in Loc and several
examples which indicate that it is unlikely that there is a characterization
similar to the characterization of regular epis in the category of topological
spaces as quotient maps.
Date: Wed, 30 Oct 1996 13:46:57 0400 (AST)
Subject: Preprints on axiomatic and synthetic domain theory.
Date: Wed, 30 Oct 1996 16:45:09 GMT
From: Marcelo Fiore
The following papers are available from http://www.dcs.ed.ac.uk/home/mf/
===============================================================================
File: SDTmodels.dvi or .ps
Two Models of Synthetic Domain Theory (by Fiore and Rosolini)
Two models of synthetic domain theory encompassing traditional categories of
domains are introduced. First, we present a Grothendieck topos embedding the
category \omegaCpo of \omegacomplete posets and \omegacontinuous functions
as a reflective exponential ideal. Second, we obtain analogous results with
respect to a category of domains and stable functions.
===============================================================================
File: ADT_and_SDT.dvi or .ps
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic
Domain Theory (by Fiore and Plotkin)
We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain
Theory (SDT). On the one hand, we introduce a class of nonelementary models
of SDT and show that the domains in them yield models of ADT. On the other
hand, for each model of ADT in a wide class we construct a model of SDT such
that the domains in it provide a model of ADT which conservatively extends the
original model.
===============================================================================
Date: Wed, 30 Oct 1996 13:46:10 0400 (AST)
Subject: Preprint (announcement)
Date: Wed, 30 Oct 1996 13:57:32 +0100
From: Anders Kock
Anders Kock: "The maximal atlas of a foliation".
This is my contribution to the 62. PSSL that has just been held. It
corrects the attempt I made in 1987 (in "Generalized Fibre Bundles") to
describe the holonomy groupoid of a foliation F in canonical terms, using
the pregroupoid structure on the maximal atlas for F (meaning the set of
all Fdistinguished germs).
The preprint is at
ftp://ftp.mi.aau.dk/pub/kock/atlas.ps
and the size is about 120 kb.
Date: Fri, 13 Dec 1996 16:06:58 0400 (AST)
Subject: preprint available
Date: Fri, 13 Dec 1996 10:46:36 +0100
From: Marco Grandis
The following hardcopy preprint is available
M. Grandis, Variables and weak limits in categories and homotopy
categories, Dip. Mat. Univ. Genova, Preprint 329 (1996).
Regards,
Marco Grandis
Abstract. Variables in a category X are introduced, extending subobjects.
Variables are well related to weak limits, as subobjects to limits; and
they may be viewed as a replacement of subobjects in categories just
possessing weak limits, typically homotopy categories.
From a formal point of view, the Freyd embedding X > FrX
(introduced to embed the stable homotopy category of spaces into an abelian
category, in Freyd, "Stable homotopy", La Jolla) allows one to reduce
variables in X to distinguished subobjects in FrX (with respect to a
canonical factorisation structure) and, loosely speaking, weak limits to
limits. Thus, "homotopy variables" for a space X, with respect to the
homotopy category HoTop, form a lattice Fib(X) of types of fibrations
over X, which can be identified to the lattice of distinguished
subobjects of X in Fr(HoTop).
Concretely, we give here various instances of the classification of
variables within finitely generated abelian groups, as a first step towards
a general classification of such variables, and of homotopy variables for
spaces having the homotopy type of a CWcomplex.
[From the Introduction:
A (categorical) *variable* of an object A is an equivalenceclass of
morphisms with values in A, where x: X  > A corresponds to
y: Y  > A iff there exist maps u, v such that x = yu, y = xv.
Among them, the *monic* variables (having some representative which
is so) can be identified to subobjects. As a motivation for the name, a
morphism x: X > A is commonly viewed within category theory as a
"variable element" of A, parametrised over X.]
Date: Sun, 2 Feb 1997 14:03:15 0400 (AST)
Subject: Manuscript
Date: Sun, 2 Feb 1997 17:11:05 GMT
From: MHEBERT@acs.auc.eun.eg
Here is an Abstract of a manuscript recently submitted, and available on
request. Part of it was presented at the last summer Sussex Category Meeting.
On generation and implicit partial operations in
locally presentable categories
Michel Hebert (The American University in Cairo, Cairo, Egypt)
Abstract. In a locally apresentable category C, seen as a category of aary
Ssorted structures, we describe the subobjects (resp. the regular, strong
subobjects) generated by a subset, first in terms of closure under specific
types of implicit partial operations (IPO), and then in syntactic terms,
using variations on the concept of dominion. This extends previous results
from [Hebert, Can. J.Math 93]. The domain of definition of an IPO of arity
s >s is a subfunctor V >> U(s) of the appropriate forgetful functor,
and each limitclosed domain V determines, in a natural way, a structure P(V)
in C having as its elements of sort s the (s>s)ary IPO's with domain V
(This generalizes the fact that the elements of sort s of the free structure F(s)
can be seen as the (s>s)ary implicit total operations in C). The P(V)'s
for subobjectclosed V >> Us with  s  < a are precisely the
agenerated objects (in the sense of GabrielUlmer) of C. Finally we use
IPO's to give a characterization of the socalled aretractions,
which parallels the known syntactic characterization of apure morphisms.
The point of view adopted in this paper is the one of the algebraist or
modeltheorist wishing to use the tools of category theory without making
radical changes in the concrete description of his/her favourite structures
(in particular without modifying the type). A part of the paper deals with
the translation problems which arise.
Date: Wed, 19 Feb 1997 11:52:46 0400 (AST)
Subject: revised paper available
Date: Wed, 19 Feb 1997 12:17:32 +0100 (MET)
From: koslowj@iti.cs.tubs.de
Hello,
A revised version of my article "A convenient category for games and
interaction" is available from my home page
http://www.iti.cs.tubs.de/TIINFO/koslowj/koslowski.html
It better substantiates my claim of last year's workshop Domains II
here in Braunschweig that the composition of games I introduced is
orthogonal to the established composition of strategies. The abstract
is appended at the end.
If you had trouble in the past reaching my home page, we did find a
faulty entry in a name server last Fall. If the problems persist,
please let me know!
 J"urgen
%% Abstract for: A convenient category for games and interaction
Guided by the familiar construction of the category rel of
relations, we first construct an orderenriched category gam .
Objects are sets, and 1cells are games, viewed as special kinds of
trees. The quest for identities for the composition of arbitrary
trees naturally suggests alternating trees of a specific
orientation. Disjoint union of sets induces a tensor product
$\otimes$ and an operation o on gam that allow us to
recover the monoidal closed category of games and strategies of
interest in game theory. Since gam does not have enough maps,
\ie, left adjoint 1cells, these operations do not have nice
intrinsic descriptions in gam . This leads us to consider games
with explicit delay moves. To obtain the ``projection'' maps
lacking in gam , we consider the Kleislicategory K induced by
the functor _+1 on the category of maps in gam . Then we
extend gam as to have K as category of maps. Now a
satisfactory intrinsic description of the tensor product exists,
which also allows us to express o in terms of simpler
operations. This construction makes clear why $\multimap$, the key
to the notion of strategy, cannot be functorial on gam .
Nevertheless, the composition of games may be viewed as orthogonal to
the familiar composition of strategies in a common framework.

J"urgen Koslowski % If I don't see you no more in this world
ITI % I meet you in the next world
TU Braunschweig % and don't be late!
koslowj@iti.cs.tubs.de % Jimi Hendrix (Voodoo Child)
Date: Fri, 21 Feb 1997 12:28:11 0400 (AST)
Subject: Locales as "topologyfree spaces"
Date: Fri, 21 Feb 1997 11:34:36 +0000
From: Steve Vickers
I am making two papers available as Departmental Research Reports:
"Topical Categories of Domains"
"Localic Completion of Quasimetric Spaces"
Both explore the idea of locales (and, indeed, toposes) as "topologyfree
spaces". The technique is to work not with frames (pointfree topologies)
but with presentations of them, understood as propositional geometric
theories whose models are the points. (But it is normally more convenient
to work with equivalent predicate theories.) Then 
* the geometric theory already determines an implicit topology on its models;
* any construction of models of one theory out of models of another
automatically determines a continuous map (or geometric morphism), just so
long as the construction is geometric.
In effect, a restriction to geometric mathematics removes the need to treat
topology explicitly, hence "topologyfree spaces". Apparently, explicit
topology is needed to correct the overcredulousness of classical reasoning
principles, though in practice the geometric constraints often end up
forcing one to reintroduce the normal topological arguments in a different
guise.
The two papers test the applicability of the idea in the two areas of
domain theory and quasimetric spaces. Aside from the "topologyfree space"
aspects, the papers develop some new results:
"Topical Categories of Domains" addresses categorical domain theory and
replaces the usual classes of objects and morphisms by toposes classifying
them. New general results concerning fixpoints of endogeometricmorphisms
of toposes exploit their intrinsically topological nature to give a simple
approach to the solution of domain equations. The paper also gives a
summary of the constructive theory of Kuratowski finite sets and
establishes some limitations to the Cartesian closedness of Sets.
"Localic Completion of Quasimetric Spaces" proposes a construction of
locales in completion of quasimetric spaces (using ideas of flatness
deriving from Lawvere's enriched category account), studies the
powerlocales and shows that a limit map from a locale of Cauchy sequences
to the completion is triquotient in the sense of Plewe.
Paper copies are available from me; electronic copies are expected to be
available shortly in the Department of Computing's Research Report series
coordinated by Frank Kriwaczek (frk@doc.ic.ac.uk).
Steve Vickers.
Date: Thu, 10 Apr 1997 16:34:46 0300 (ADT)
Subject: Preprint available.
The following preprint is available at
http://www.dcs.ed.ac.uk/home/mf/ADT/
as cub.dvi and cub.ps. Best, Marcelo.
Complete Cuboidal Sets in Axiomatic Domain Theory
Marcelo Fiore Gordon Plotkin John Power
Department of Computer Science
Laboratory for Foundations of Computer Science
University of Edinburgh, The King's Buildings
Edinburgh EH9 3JZ, Scotland
Synopsis
We study the enrichment of models of axiomatic domain theory. To this end, we
introduce a new and broader notion of domain, viz. that of complete cuboidal
set, that complies with the axiomatic requirements. We show that the category
of complete cuboidal sets provides a general notion of enrichment for a wide
class of axiomatic domaintheoretic structures.
Cuboidal sets play a role similar to that played by posets in the traditional
setting. They are the analogue of simplicial sets but with the simplicial
category enlarged to the cuboidal category of cuboids, i.e. of finite
products O_n1 x ... x O_ni of finite ordinals. These cuboids are the
possible shapes of paths. A cuboidal set P has a set P(C) of paths of
every shape C = n1 x ... x ni; indeed, it is a (rooted) presheaf over the
cuboidal category. The set of points of P is P(O_1). The set of
(onedimensional) paths of length n is P(O_n+1); they can be thought of
as (linear) computations conditional on the occurrence of n linearly
ordered events e_1 < ... < e_n. Evidently, O_n is the partial order
associated to this simple linear event structure, and can be considered as a
sequential process of length n. At higher dimensions, P(O_n1 x ... x O_ni)
can be thought of as the set of computations conditional on the occurrence of
n_1 + ... + n_i events ordered by e_1,1 < ... < e_1,n1 ; ... ;
e_i,1 < ... < e_i,ni. This is the event structure which can be considered
as i sequential processes, of respective lengths O_n1, ..., O_ni, running
concurrently.
Complete cuboidal sets are cuboidal sets equipped with a formallub operator
satisfying three algebraic laws, which are exactly those needed of the lub
operator in order to prove the fixedpoint theorem. Computationally, the
passage from cuboidal sets to complete cuboidal sets corresponds to allowing
infinite processes. In fact, the formallub operator assigns paths of shape
C to `paths of shape C x omega', for every C. Here the set of paths of
shape C x omega is the colimit of the paths of shape C x O_n; such paths
can be thought of as the higherdimensional analogue of the increasing
sequences of traditional domain theory.
Date: Sat, 12 Apr 1997 16:01:26 0300 (ADT)
Subject: new preprint available
Date: Sat, 12 Apr 97 16:40:29 +1000
From: Max Kelly
A LaTeX preprint of the paper "On the monadicity over graphs of
categories with limits", by G.M. Kelly and I.J. Le Creurer, and to
appear in Cahiers de Topologie et Ge'om. Diff. Cate'goriques, is
in our public site sydcat at maths.usyd.edu.au (=129.78.68.2),
in the directory sydcat/papers/kelly, as the file named
monograph.tex .
I presume I already announced another comparativelyrecent preprint
there: to wit, the file jk.tex contains "The Reflectiveness of
Covering Morphisms in Algebra and Geometry", by G. Janelidze and
G.M. Kelly, which is still with the referee.
Max Kelly.
Date: Wed, 16 Apr 1997 17:21:16 0300 (ADT)
Subject: protected files
Date: Mon, 14 Apr 97 15:20:49 +1000
From: Max Kelly
Several readers have pointed out that the new files monograph.tex and
jk.tex, whose availability on our site sydcat I announced the other day
were protected; this oversight has now been corrected, Max Kelly.
Date: Thu, 24 Apr 1997 15:26:59 0300 (ADT)
Subject: preprint available
Date: Tue, 22 Apr 1997 21:26:04 +0200 (METDST)
From: Anders Kock
The article:
"Geometric Construction of the LeviCivita Parallelism"
by Anders Kock
is available from
ftp://ftp.mi.aau.dk/pub/kock/parallel.ps
(about 150 kb).
(The LeviCivita Parallellism is also called the Riemannian Connection; it
is the unique symmetric affine connection compatible with a given
Riemannian metric. We present a geometric construction of it, using
variational principles and synthetic differential geometry.)
Date: Thu, 24 Apr 1997 15:29:45 0300 (ADT)
Subject: Notes of two lectures
Date: Thu, 24 Apr 1997 15:37:19 +1100
From: Ross Street
This is to announce the placement on the WWW of the notes of my two
lectures at the Conference on Higher Category Theory and Mathematical
Physics, Northwestern University (Evanston, Illinois; 2830 March 1997).
The site is:
[I have tried to eliminate offending fonts and to accommodate funny US
paper size. Thanks to Sjoerd Crans for helping here.]
Title: The role of Michael Batanin's monoidal globular categories
Lecture I: Globular categories and trees
Lecture II: Higher operads and weak omegacategories
This is a report on recent work of Michael Batanin. The goal of his
work is to provide an environment for defining the concepts associated with
weak omegacategories and for developing the ensuing theory. The approach
is "globular".
To put this in context, I might mention some important steps in the
development of weak omegacategories. Categories were defined by
EilenbergMac Lane in 1945. Monoidal and symmetric monoidal categories were
defined by Mac Lane in 1963. Ehresmann defined (strict) ncategories in
1966. Bénabou defined bicategories in 1967. In the early 80s, monoidal
bicategories were in the air but a full definition was not published in
that period. JoyalStreet defined braided monoidal categories in 1985.
GordonPowerStreet defined tricategories in 1991 (this, and the coherence
theorem, were published in 1995). Braided monoidal categories were defined
by KapranovVoevodskyBaezNeuchlBreen around 1993. Trimble produced a
definition of tetracategory in 1995.
Diverse approaches to weak ncategories for all n have appeared.
Street (1985) suggested a simplicial definition with horn filler
conditions. Trimble (1994) approached the problem using operads and
Stasheff associahedra. BaezDolan (1995) have a definition using typed
operads and opetopes. Tamsamani (1996) gave a multisimplicial definition.
Batanin uses higher operads and globular sets.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Ross Street email: street@mpce.mq.edu.au
Mathematics Department phone: +612 9850 8921
Macquarie University fax: +612 9850 8114
Sydney, NSW 2109
Australia Internet: http://www.mpce.mq.edu.au/~street/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Date: Mon, 28 Apr 1997 07:29:10 0300 (ADT)
Subject: weak \omegacategories
Date: Fri, 25 Apr 1997 14:56:43 +1100
From: Olga Batanin
The preprint version of my paper
"Monoidal globular categories as a natural environment for the theory
of weak ncategories"
is now available. The dvi file is at
http://wwwmath.mpce.mq.edu.au/~mbatanin/coh0.dvi
Please, contact me if you have any difficulties with printing it out.
I can mail hard copies.
Michael Batanin.
Abstract.
The paper is devoted to the problem of defining weak
$\omega$categories.
The definition presented here is based on a nontrivial
generalization of the apparatus of operads and their algebras,
originally developed by P.May \cite{May} for the needs of algebraic
topology.
Yet, for the purposes of higher order category theory, a higher
dimensional notion of operad is required.
Briefly, the idea of a higher operad may be explained as follows.
An ordinary nonsymmetric operad in $Set$ associates a set $A_{n}$
to every integer $n$. The set of integers may be interpreted
as the set of $1$cells in the free category generated by one object
and one nonidentity endomorphism of this object. To find a higher order
generalization of the notion of operad we have to describe the free
strict $\omega$category generated by one object
and one nonidentity endomorphism of this object and one nonidentity
endomorphism of this endomorphism and so on (so, for example, the set
of integers is the onedimensional part of this category). The required
$\omega$category $Tr$ will be the category of planar trees of a special
type. The $k$th composition of cells will be given by the colimit of
the diagram of trees over a special tree $M_{n}^{k}$.
The other component of the theory of operads is an appropriate
monoidal category (with some extrastructure like braiding
or symmetry) where one can consider the notion of operad. We
need also a monoidal category (perhaps, with extrastructure as
well) where one can define the notion of algebra for an operad. Finally,
the corresponding coherence theorems for both types of monoidal
categories are required.
I call all these components a natural environment
for a given theory of operads. One of my main goals was to find a
natural environment for the theory of higher order operads.
For this I introduce monoidal globular categories and show
they are suitable for the development of the theory of higher order
operads. The crucial point here is a coherence theorem for monoidal
globular categories (section 4) which includes as special cases the
coherence theorems for monoidal, symmetric monoidal, and braided
monoidal categories and a sort of pasting theorem for
$\omega$categories.
A primary example of a globular monoidal category is the globular
category of $n$spans $Span$. The $0$spans are just the sets. The
$1$spans are the spans in $Set$ in the usual sense. In some informal
sense, an $n$span is a relation between two $(n1)$spans. This
globular monoidal category plays the same role for higherorder
category theory as the category of sets does for ordinary category
theory.
These results allow me to formulate the notion of higher order
operad.
An $\omega$operad will associate an $n$span to
every $n$cell in $Tr$ for every $n\ge 0$.} There are
also the units and multiplications and some axioms for
these operations.
The category of nonsymmetric operads (in the
category of sets) is just a onedimensional subcategory of the category
of $\omega$operads.
Finally, Iintroduce a notion of a contractible
$\omega$operad, So the main definition is:
A weak $\omega$category is a globular set together
with the structure of algebra over a universal
contractible $\omega$operad.
I construct also a fundamental $n$groupoid functor from topological
spaces to the category of weak $n$categories for all $n$ including
$\omega$ and consider another examples of weak $n$categories, hifger
operads and their algebras.
Date: Tue, 29 Apr 1997 20:23:53 0300 (ADT)
Subject: weak \omegacategories
Date: Tue, 29 Apr 1997 23:13:53 +1000
From: Michael Batanin
Some people informed me that they have had the difficulties in printing
out the dvi file of my paper
"Monoidal globular categories as a natural environment for the theory
of weak ncategories"
It seems that the following address works better
http://wwwmath.mpce.mq.edu.au/~mbatanin/papers.html
You can find here the .ps file of my paper.
Michael Batanin.
Date: Wed, 30 Apr 1997 20:00:52 0300 (ADT)
Subject: Notes of two lectures
Date: Wed, 30 Apr 1997 11:39:29 +1100
From: Ross Street
With computers, things we expect to be simple never are!
Last week I announced the placement on the WWW of the notes (see the short
description below) of my two lectures at the Conference on Higher Category
Theory and Mathematical Physics, Northwestern University (Evanston,
Illinois; 2830 March 1997). The site is:
I'll spare you the details of the problems, but, up until today, what was
at this site was an old version prepared before the conference. This is not
what I had intended. I am truly sorry to people who have downloaded that
version already.
The correct version is NOW at the site. There still seems to be a problem
when the document is viewed (by some "ghost" technology) but it does (at
least for us) print out pretty well. There is a colour table on page 2
which even seems to view correctly! I am extremely grateful to Ross Moore
and Sjoerd Crans for helping me out of the mess I (and my little Mac)
created.
I invite people who have downloaded the old version to try again; I am
really sorry.
Title: The role of Michael Batanin's monoidal globular categories
Lecture I: Globular categories and trees
Lecture II: Higher operads and weak omegacategories
This is a report on recent work of Michael Batanin. The goal of his
work is to provide an environment for defining the concepts associated with
weak omegacategories and for developing the ensuing theory. The approach
is "globular". Apart from providing a precise definition of weak
omegacategory, the work gives a new algebra of planar trees and uses them
to define higher operads which I believe will find many other applications.
Note that, in the meantime (and again not without trouble!), Michael
Batanin has announced his paper containing the full details of this part of
his work. It is available at:
Happy surfing and enjoy the trees.
Ross
Date: Thu, 15 May 1997 22:19:05 0300 (ADT)
Subject: An introduction to ncategories
Date: Tue, 13 May 1997 17:29:58 0700 (PDT)
From: john baez
Here is the abstract of a paper that is now available in Postscript
form at:
http://math.ucr.edu/home/baez/ncat.ps
If downloading it or printing it out is a problem, I can mail copies
to people.

An Introduction to nCategories
John C. Baez
An ncategory is some sort of algebraic structure consisting of objects,
morphisms between objects, 2morphisms between morphisms, and so on up
to nmorphisms, together with various ways of composing them. We survey
various concepts of ncategory, with an emphasis on `weak' ncategories,
in which all rules governing the composition of jmorphisms hold only up
to equivalence. (An nmorphism is an equivalence if it is invertible,
while a jmorphism for j < n is an equivalence if it is invertible up to
a (j+1)morphism that is an equivalence.) We discuss applications of
weak ncategories to various subjects including homotopy theory and
topological quantum field theory, and review the definition of weak
ncategory recently proposed by Dolan and the author.
Date: Tue, 3 Jun 1997 11:06:15 0300 (ADT)
Subject: New versions of papers available
Date: Mon, 02 Jun 1997 15:18:51 0400
From: Charles Wells
New versions of three papers by Atish Bagchi and myself are available at
http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/pub/papers.html
The papers are
Varieties of Mathematical Prose
Graph Based Logic and Sketches I: The General Framework
Graph Based Logic and Sketches II: Finite Product Categories and Equational
Logic
Charles Wells, 105 South Cedar Street, Oberlin, Ohio 44074, USA.
EMAIL: cfw2@po.cwru.edu.
HOME PHONE: 216 774 1926. FAX: Same as home phone.
HOME PAGE: URL http://www.cwru.edu/CWRU/Dept/Artsci/math/wells/home.html
"Some have said that I can't sing. But no one will say that I _didn't_ sing."
Florence Foster Jenkins
Date: Wed, 11 Jun 1997 11:53:50 0300 (ADT)
Subject: Announcement: paper on linear functors available
Date: Wed, 11 Jun 1997 10:13:43 0400
From: Robert A. G. Seely
We wish to announce the availability of the following paper.
Linearly distributive functors
by
J.R.B. Cockett
R.A.G. Seely
ABSTRACT
This paper introduces a notion of "linear functor" between linearly
distributive categories that is general enough to account for common
structure in linear logic, such as the exponentials (!, ?), and the
additives (product, coproduct), and yet when interpreted in the doctrine of
*autonomous categories, gives the familiar notion of monoidal functor. We
show that there is a biadjunction between the 2categories of linearly
distributive categories and linear functors, and of *autonomous categories
and monoidal functors, given by the construction of the "nucleus" of a
linearly distributive category. We develop a calculus of proof nets for
linear functors, and show how linearity accounts for the essential structure
of the exponentials and the additives.
This paper was first presented at a conference held in Montreal in May 1997,
in honour of Michael Barr's 60th birthday, and is dedicated to him in
celebration of this occasion.

The paper may be found at the following URLs
or from the WWW home page
Contact if there is any problem retrieving this paper.
Date: Tue, 1 Jul 1997 15:12:45 0300 (ADT)
Subject: Preprint available
Date: Tue, 1 Jul 97 13:38 BST
From: Dr. P.T. Johnstone
The Pure Mathematics Department of Cambridge University has a new electronic
preprint server (accessible via our home page at http://www.pmms.cam.ac.uk).
The first preprint available may be of interest to people on the categories
mailing list: it is
C. Butz and P.T. Johnstone: Classifying toposes for firstorder theories
Abstract: By a classifying topos for a firstorder theory $\Bbb T$, we mean
a topos $\cal E$ such that, for any topos $\cal F$, models of $\Bbb T$ in
$\cal F$ correspond exactly to open geometric morphisms ${\cal F}
\rightarrow{\cal E}$. We show that not every (infinitary) firstorder theory
has a classifying topos in this sense, but we characterize those which do by
an appropriate `smallness condition', and we show that every Grothendieck
topos arises as the classifying topos of such a theory. We also show that
every firstorder theory has a conservative extension to one which possesses
a classifying topos, and we obtain a Heytingvalued completeness theorem for
infinitary firstorder logic.
For those who would prefer to receive a hard copy of this paper, I shall be
bringing a supply with me to the Vancouver meeting.
Peter Johnstone
Date: Thu, 3 Jul 1997 22:13:34 0300 (ADT)
Subject: announcement
Date: Thu, 3 Jul 1997 20:52:47 0400 (EDT)
From: Michael Makkai
The following paper is announced:
On weak higher dimensional categories
by Claudio Hermida, Michael Makkai and John Power
Abstract:
Inspired by the concept of opetopic set introduced in a recent paper by
John C. Baez and James Dolan, we give a modified notion called multitopic
set. The name reflects the fact that, whereas the Baez/Dolan concept is
based on operads, the one in this paper is based on multicategories. The
concept of multicategory used here is a mild generalization of the
samenamed notion introduced by Joachim Lambek in 1969. Opetopic sets and
multitopic sets are both intended as vehicles for concepts of weak higher
dimensional category. Baez and Dolan define weak ncategories as
(n+1)dimensional opetopic sets satisfying certain properties. The version
intended here, multitopic ncategory, is similarly related to multitopic
sets. Multitopic ncategories are not described in the present paper; they
are to follow in a sequel. The present paper gives complete details of the
definitions and basic properties of the concepts involved in multitopic
sets. The category of multitopes, analogs of opetopes of Baez and Dolan,
is presented in full, and it is shown that the category of multitopic sets
is equivalent to the category of setvalued functors on the category of
multitopes.
The paper is available by anonymous ftp from triples.math.mcgill.ca in
directory pub/makkai, or via the CRTC home page
ftp://triples.math.mcgill.ca/crtc.html
and click on "Makkai". The paper is in nine files, each with a name
starting with `mult`; they are PostScript files.
I will send a limited number of hard copies upon request.
Michael Makkai
Date: Tue, 8 Jul 1997 14:20:18 0300 (ADT)
Subject: announcement of preprint
Date: Tue, 8 Jul 1997 16:32:22 +0200
From: I. Moerdijk
Dear colleagues,
A psfile of the following short preprint can be picked up from my homepage
(http://www.math.ruu.nl/people/moerdijk)
I Moerdijk, J. Vermeulen, Proof of a conjecture of A. Pitts.
Abstract: Using only elementary properties of inverse limits and localization,
we prove the BeckChevalley condition for lax pullbacks of coherent toposes. In
this way, we obtain a simple and constructive proof of the descent theorem for
coherent (pre)toposes.
With best regards, Ieke Moerdijk.
Date: Tue, 8 Jul 1997 14:18:24 0300 (ADT)
Subject: addendum to announcement
Date: Mon, 7 Jul 1997 18:03:49 0400 (EDT)
From: Michael Makkai
A couple of days ago I announced the paper "On weak higher dimensional
categories" by C. Hermida, M. Makkai and J. Power. Now, I am announcing
some small changes of the arrangements concerning the electronic access to
the paper.
The paper is available by anonymous ftp from triples.math.mcgill.ca in the
directory pub/makkai/multitopicsets [so, the change is that now the paper
is put into a subdirectory of pub/makkai]. There are ten files [there were
nine before; I have cut the largest into two].
M. Makkai
Date: Thu, 10 Jul 1997 16:51:01 +0200 (MET DST)
Subject: un peu de r'eclame
Date: Wed, 9 Jul 1997 18:01:19 +0200
From: Pierre Ageron
Let me advertise for some more or less recent work of mine
about sketches/accessible categories (not available electronically,
but I'll be happy to send reprints or preprints on request).

(1) Cat'egories accessibles `a limites projectives non vides et
cat'egories accessibles `a limites projectives finies
Diagrammes 34 (1995) 110
For fixed b, baccessible categories with nonempty limits are
characterized as the categories of models of specific sketches.
As a corollary, the category of these categories is Cartesian closed.
(Proved independantly by Ad'amek.)
Accessible categories with finite limits are also characterized.

(2) Effective taxonomies and crossed taxonomies
Cahiers de Top. et de G'eom. Diff. Cat. XXXVII (1996) 8290
A taxonomy is a "category without identities". This bare structure is
somewhat dull, but "crossed modules of taxonomies" seem more interesting.
In the latter structure, "Dedekindfinite" objects play a role
similar to that of finitely presentable objects in a category.
A notion similar to that of accessibility can thus be defined.

(3) La tour holomorphe d'une esquisse
Cahiers de Top. et de G'eom. Diff. Cat. XXXVII (1996) 295314
A construction of Lair's in the category of sketches is revisited
and noticed to specialize to the construction of the holomorph
when restricted to groups. The iteration of this construction
reveals two invariants of a sketch: an ordinal and a group.
Some explicit computations are provided.

(4) Cat'egories accessibles `a produits fibr'es
(preprint)
Continuation of (1). Accessible categories with (finite) pullbacks
are characterized in terms of sketches. This is achieved
by introducing "free" colimits in Set: such colimits
are proved to be exactly those that commute with pullbacks.

(5) Limites projectives conditionnelles dans les cat'egories accessibles
(preprint)
For fixed b, those baccessible categories s.t. every diagram with a
cone has a limit are characterized in terms of sketches.
As a corollary, the category of these categories is Cartesian closed.
Similarly for those baccessible categories s.t. every nonempty
diagram with a cone has a limit, or for those with
"consistent wide pullbacks".

PIERRE AGERON
1) coordonnees bureau
adresse : mathematiques, Universite de Caen, 14032 Caen Cedex
telephone : 02 31 56 57 37
telecopie : 02 31 93 02 53
adresse electronique : ageron@math.unicaen.fr
2) coordonnees domicile
adresse : 28 rue de Formigny 14000 Caen
telephone : 02 31 84 39 67
Date: Wed, 30 Jul 1997 13:39:54 0300 (ADT)
Subject: Preprint available
Date: Wed, 30 Jul 1997 15:51:32 +0200 (MET DST)
From: Carsten Butz
Dear Colleagues,
the psfile of the following preprint is available at the homepage
http://www.brics.dk/~butz :
Topological Completeness for HigherOrder Logic
by Steve Awodey (awodey@cmu.edu),
Carsten Butz (butz@brics.dk).
Abstract: Using recent results in topos theory, two systems of
higherorder logic are shown to be complete with respect to sheaf
models over topological spacessocalled ``topological semantics''.
The first is classical higherorder logic, with relational
quantification of finitely high type; the second system is a
predicative fragment thereof with quantification over functions
between types, but not over arbitrary relations. The second theorem
applies to intuitionistic as well as classical logic.
Best regards,
Steve Awodey and Carsten Butz
Date: Wed, 6 Aug 1997 08:47:28 0300 (ADT)
Subject: Preprints available
Date: Tue, 5 Aug 1997 17:19:51 0400
From: Walter Tholen
The following two preprints (joint work with George Janelidze) are available as
postscript files from my home page at
http://www.math.yorku.ca/Who/Faculty/Tholen/menu.html
For titles and abstracts, see below.
Walter Tholen

"Functorial Factorization, Wellpointedness and Separabilty"
Abstract: A functorial treatment of factorization structures is presented,
under extensive use of wellpointed endofunctors. Actually, socalled weak
factorization systems are interpreted as pointed lax indexed endofunctors,
and this sheds new light on the correspondence between reflective subcategories
and factorization systems. The second part of the paper presents two improtant
factorization structures in the context of pointed endofunctors:
concordantdissonant and inseparablesepaprable.
"Extended Galois Theory And Dissonant Morphisms"
Abstract: For a given Galois structure on a category C and an effective descent
morphism p: E > B in C we describe the category of socalled weakly split
objects over (E,p) in terms of internal actions of the Galois (pre)groupoid of
(E,p) with an additional structure. We explain that this generates various
known results in categorical Galois theory and in particular two results of M.
Barr and R. Diaconescu. We also give an elaborate list of examples and
applications.
Date: Thu, 7 Aug 1997 14:13:07 0300 (ADT)
Subject: preprint
Date: Thu, 7 Aug 1997 15:43:23 +1000
From: Michael Batanin
The preprint
" Finitary monads on globular sets and notions of computad they generate "
is available as
postscript files at
http://wwwmath.mpce.mq.edu.au/~mbatanin/papers.html
Abstract
Consider a finitary monad on the category of globular sets. We prove
that the category of its algebras is isomorphic to the category
of algebras of an appropriate monad on the
special category (of computads) constructed from the data of the
initial monad. In the case of the free $n$category monad this
definition coincides with R.Street's definition of $n$computad. In
the case of a monad generated by a higher operad this allows us to
define a pasting operation in a weak $n$category. It may be also considered
as the first step toward the proof of equivalence of the different
definitions of weak $n$categories.
Date: Tue, 2 Sep 1997 09:19:27 0300 (ADT)
Subject: preprint available
Date: Mon, 1 Sep 1997 14:47:02 +0200 (MET DST)
From: Koslowski
Dear Colleagues,
An updated preprint of my paper "Beyond the Chuconstruction", which
I presented in Vancouver in July, is now available on my webpage
http://www.iti.cs.tubs.de/TIINFO/koslowj/koslowski.html
The abstract follows below.
From a symmetric monoidal closed (= autonomous) category PoHsiang
Chu originally constructed a *autonomous one, ie, a selfdual
autonomous category where the duality is realized by means of a
dualizing object. Recently, Michael Barr introduced an extension
for the nonsymmetric, but closed, case that after an initial step
utilized monads and modules between them. Since these tools are
wellunderstood in a bicategorical setting, we introduce a notion
of local *autonomy for closed bicategories that turns out to
be inherited by the bicategories of monads and the bicategory of
interpolads. Since the first step of Barr's construction carries
over directly to the bicategorical setting, we recover his main
result as an easy corollary. Furthermore, the Chuconstruction at
this level may be viewed as a procedure to turn the endo1cells of
a bicategory into the objects of a new bicategory, and hence is
conceptually close to the constructions of bicategories of monads
and of interpolads.
Best regards,
 J"urgen

J"urgen Koslowski % If I don't see you no more in this world
ITI % I meet you in the next world
TU Braunschweig % and don't be late!
koslowj@iti.cs.tubs.de % Jimi Hendrix (Voodoo Child)
Date: Wed, 3 Sep 1997 15:29:23 0300 (ADT)
Subject: preprint available: not yet :(
Date: Tue, 2 Sep 1997 15:19:46 +0200 (MET DST)
From: koslowj@iti.cs.tubs.de
Dear Colleagues,
A technical glitch prevents me from puttting the preprint "Beyond the
ChuConstruction", announced yesterday, on my web page ("no space on
device", which doesn't make sense).
I won't be able to fix this until I return from CTCS97 next Sunday,
since our technician isn't here right now. Please wait until next
Monday or Tuesday before downloading the paper.
I'm sorry about this problem.
 J"urgen

J"urgen Koslowski % If I don't see you no more in this world
ITI % I meet you in the next world
TU Braunschweig % and don't be late!
koslowj@iti.cs.tubs.de % Jimi Hendrix (Voodoo Child)
Date: Mon, 8 Sep 1997 15:07:16 0300 (ADT)
Subject: For Category Bulletin: New Preprint
Date: Mon, 8 Sep 1997 16:27:35 +0100 (BST)
From: Ronnie Brown
Groupoids and Crossed Objects in Algebraic Topology
Ronald Brown
Notes for Lectures at the Summer School on the `Foundations of Algebraic
Topology', Grenoble, June 14 July 5, 1997 (71 pages).
Abstract:
The notes concentrate on the background, intuition, proof and
applications of the 2dimensional Van Kampen Theorem
(for the fundamental crossed module of a pair), with
sketches of extensions to higher dimensions.
One of the points stressed is how the extension from
groups to groupoids leads to an extension from the abelian
homotopy groups to non abelian higher dimensional
generalisations of the fundamental group, as was sought
by the topologists of the early part of this century.
This links with J.H.C. Whitehead's efforts to extend
combinatorial group theory to higher dimensions
in terms of combinatorial homotopy theory, and which
analogously motivated his simple homotopy theory.
Available from
http://www.bangor.ac.uk/~mas010/brownpr.html
(gzipped postscript).
Ronnie Brown
Prof R. Brown, School of Mathematics,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474office: 382475
fax: +44 1248 383663
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
New article: Higher dimensional group theory
Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm
Date: Thu, 2 Oct 1997 16:47:17 0300 (ADT)
Subject: resolutions as fractions, space complexity
Date: Thu, 2 Oct 1997 12:19:44 0500 (CDT)
From: J. R. Otto
Dear People,
The following revisions of talks on work in progress may be of
interest.
From NNO to Complexity (12 pages, October 2, 1997). We
begin to revisit space complexity by collapsing resolutions to maps.
So we evolve our talk `Presenting LCC Categories by Answering Queries'
by stratifying higher order types and allowing alternatives.
Presenting LCC Categories by Answering Queries (15
pages, October 1, 1997). We present LCC categories in a manner that
provides a basis for logic programming with dependent types and
equality. We find that resolutions are left fractions which collapse
to the maps.
They are linked to http://www.mcs.net/~quant/ .
Regards, Jim Otto
quant@mcs.com
Date: Mon, 13 Oct 1997 09:17:41 0300 (ADT)
Subject: Operads, multicategories
Date: Mon, 13 Oct 1997 10:36:43 +0100 (BST)
From: Tom Leinster
An advertisement for an article, available by electric transmission from
http://www.dpmms.cam.ac.uk/~leinster.
ABSTRACT
Notions of `operad' and `multicategory' abound. This work provides a single
framework in which many of these various notions can be expressed.
Explicitly: given a monad * on a category S, we define the term
(S,*)multicategory, subject to certain conditions on S and *. Different
choices of S and * give some of the existing notions. We then describe the
algebras for an (S,*)multicategory, and finish with a selection of possible
further developments. Our approach enable concise descriptions of Baez and
Dolan's opetopes and Batanin's operads; both of these are included.
Tom Leinster
Date: Thu, 16 Oct 1997 16:53:03 0300 (ADT)
Subject: Weak higher dimensional categories
Date: Thu, 16 Oct 1997 10:13:07 +0100
From: ajp@dcs.ed.ac.uk
Those people interested in Tom Leinster's paper on multicategories and
weak higher dimensional categories might also be interested in recent,
closely related work by Claudio Hermida at McGill. I do not think
there is a paper available yet, but he has given talks at Vancouver
and at the recent meeting of the Canadian Math Society in Montreal, so
there are probably slides available.
Date: Sun, 9 Nov 1997 14:21:08 0400 (AST)
Subject: higherdimensional multicategories and weak ncategories
Date: Fri, 7 Nov 1997 17:45:34 +0000 (GMT)
From: Claudio Hermida
Dear all,
I've just finished scanning the slides of my talks at CT97 (Vancouver) and
the AMS meeting at Montreal on
Higherdimensional multicategories
aimed as a set up for weak ncategories in the Baez/Dolan sense, ie. using
universally defined composites to avoid coherence conditions.
The slides are a series of .JPG files accessible through my web page
http://www.math.mcgill.ca/~hermida
Claudio Hermida
Date: Thu, 13 Nov 1997 15:56:31 0400 (AST)
Subject: preprints available
Date: Wed, 12 Nov 1997 14:16:30 +0100 (MET)
From: Anders Kock
The following preprints are available:
Differential Forms as Infinitesimal Cochains
This is essentially my contribution at the Vancvouver Category Theory
Meeting in July. It proves that the simplicial complex given by the first
neighbourhood of the diagonal of a manifold (in a well adapted model for
SDG) has de Rham cohomology of the manifold as its Rdual.
Extension Theory for Local Groupoids
We relate Extension Theory for (nonabelian) groups (a la EilenbergMac
Lane) with the theory of Connections (a la Ehresmann), via a notion of
local groupoid. In particular, we give in this setting a kind of converse
to the statement "the curvature 2form of a connection satisfies Bianchi
identity".
Both these preprints are accessible via my home page:
http://www.mi.aau.dk/~kock/
or directly at
ftp://ftp.mi.aau.dk/pub/kock/Cochains.ps
(respectively ../locg.ps)
Anders Kock
Date: Thu, 13 Nov 1997 15:57:52 0400 (AST)
Subject: Limits in double categories, preprint
Date: Wed, 12 Nov 1997 19:34:32 +0100
From: Marco Grandis
The following preprint is available:
Limits in double categories
by Marco Grandis and Robert Pare
Abstract. We define the notion of (horizontal) double limit for a double
functor F: I > A between double categories, and we give a construction
theorem for such limits, from double products, double equalisers and
tabulators (the double limits of vertical arrows). Double limits can
describe important tools; for instance, the Grothendieck construction of a
profunctor is its tabulator, in the "double category" of categories,
functors and profunctors. If A is a 2category, the previous result
reduces to Street's construction theorem of weighted limits, by ordinary
limits and cotensors 2*X (the tabulator of the vertical identity of the
object X).
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
email: grandis@dima.unige.it
tel: +39.10.353 6805 fax: +39.10.353 6752
home page: http://www.dima.unige.it/STAFF/GRANDIS/
Date: Thu, 27 Nov 1997 16:02:23 0400 (AST)
Subject: Big Omega available
Date: Thu, 27 Nov 1997 10:31:13 +1100
From: Ross Street
Dear Colleagues
This is to announce the availability at
http://wwwmath.mpce.mq.edu.au/~mbatanin/Bigomega.ps
of a short 10 page note which begins as follows:
**********************************************************************
"The universal property of the multitude of trees"
Michael Batanin and Ross Street
Macquarie University, N S W 2109
AUSTRALIA
Email and
November 1997
Lawvere [BL] essentially pointed out that the category Delta,
whose objects are finite ordinals and whose arrows are orderpreserving
functions, is the generic monoidal category containing a monoid. Let Mon
be the category of monoids in the category Set of sets. Bénabou [Be]
pointed out that the (simplicial) nerve of the category Delta is the
standard resolution [BB] of the terminal monoid via the comonad generated
by the underlying functor Mon > Set and its left adjoint.
Let Omcat denote the category of omegacategories and let Glob
denote the category of globular sets. In this note we announce a generic
property of the category BigOmega whose nerve is the standard resolution
of the terminal omegacategory via the comonad generated by the underlying
functor Omcat > Glob and its left adjoint. We also give a concrete
model for BigOmega in terms of trees. Furthermore, we make connections
with the recent work of Joyal [J]. Full proofs of our claims will appear
elsewhere.
**************************************************************************
Regards,
Michael Batanin and Ross Street
Date: Thu, 4 Dec 1997 09:48:43 0400 (AST)
Subject: preprints available
Date: Thu, 4 Dec 1997 11:54:52 +0100
From: Marco Grandis
The following preprints are now accessible as psfiles, via web of ftp:
http://www.dima.unige.it/STAFF/GRANDIS/
ftp://www.dima.unige.it/pub/STAFF/GRANDIS
(1). "Limits in double categories", by Marco Grandis and Robert Pare
Dbl.Dec97.ps
(2). "Weak subobjects and weak limits in categories and homotopy
categories", by M.G.
Var1.Aug97.ps
(3). "Weak subobjects and the epimonic completion of a category", by M.G.
Var2.Dec97.ps
***
The first was announced on this mailing list, on 13 Nov 1997.
(With respect to the printed preprint, this is a slightly revised version,
containing a more detailed comparison with BastianiEhresmann's "limits
relative to double categories".)
The second and third form an expanded version of a printed preprint
("Variables and weak limits in categories and homotopy categories", Dec
1996), announced on this list on 13 Dec 1996.
Abstracts for (2) and (3) are given below.
***
(2). Abstract. We introduce the notion of "variation", or "weak
subobject", in a category, as an extension of the notion of subobject. The
dual notion is called a covariation, or weak quotient.
Variations are important in homotopy categories, where they are well
linked to weak limits, much in the same way as, in "ordinary" categories,
subobjects are linked to limits. Thus, "homotopy variations" for a space
S, with respect to the homotopy category HoTop, form a lattice Fib(S)
of "types of fibration" over S.
Nevertheless, the study of weak subobjects in ordinary categories, like
abelian groups or groups, is interesting in itself and relevant to classify
variations in homotopy categories of spaces, by means of homology and
homotopy functors. (To appear in: Cahiers Top. Geom. Diff. Categ.)
(3). Abstract. Formal properties of weak subobjects are considered. The
variations in a category X can be identified with the (distinguished)
subobjects in the epimonic completion of X, or Freyd completion FrX,
the free category with epimonic factorisation system over X, which
extends the Freyd embedding of the stable homotopy category of spaces in an
abelian category (P. Freyd, Stable homotopy, La Jolla 1965).
If X has products and weak equalisers, as HoTop and various other
homotopy categories, FrX is complete. If X has zeroobject, weak
kernels and weak cokernels, as the homotopy category of pointed spaces,
then FrX is a "homological" category. Finally, if X is triangulated,
FrX is abelian and the embedding X > FrX is the universal homological
functor on X, as in the original case. These facts have consequences on
the ordered sets of variations.
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
email: grandis@dima.unige.it
tel: +39.10.353 6805 fax: +39.10.353 6752
http://www.dima.unige.it/STAFF/GRANDIS/
Date: Thu, 15 Jan 1998 17:12:24 0400 (AST)
Subject: Report available
Date: Thu, 15 Jan 1998 14:37:37 +0000 (GMT)
From: Alan Jeffrey
I'd like to announce a technical report relating Power and Robinson's
premonoidal categories to a category of mixed dataflow and
controlflow graphs.
Premonoidal Categories and a Graphical View of Programs
Alan Jeffrey
University of Sussex
The report is available electronically from:
http://www.cogs.susx.ac.uk/users/alanje/premon/
Abstract

This paper describes the relationship between two different
presentations of the semantics of programs:
* Mixed data and control flow graphs are commonly used
in software engineering as a semiformal notation for describing
and analysing algorithms.
* Category theory is used as an abstract presentation of
the mathematical structures used to give a formal semantics to
programs.
In this paper, we formalize an appropriate notion of flow graph,
and show that acyclic flow graphs form the initial
symmetric premonoidal category. Thus, giving a semantics
for a programming language in flow graphs uniquely determines
a semantics in any symmetric premonoidal category.
For languages with recursive definitions, we show that cyclic flow
graphs form the initial partially traced cartesian category.
Finally, we conclude with some more speculative work, showing how
closed structure (to represent higherorder functions) or
twocategorical structure (to represent operational semantics)
might be included in this graphical framework.
The semantics has been implemented as a Java applet, which takes a
program text and draws the corresponding flow graph (all the diagrams
in this paper are drawn using this applet).
The categorical presentation is based on Power and Robinson's
premonoidal categories and Joyal, Street and Verity's monoidal traced
categories, and uses similar techniques to Hasegawa's semantics for
recursive declarations. The closed and twocategorical structure is
related to Gardner's namefree presentation of Milner's action
calculi.
Date: Mon, 19 Jan 1998 09:45:42 0400 (AST)
Subject: announcement
Date: Sun, 18 Jan 1998 18:33:38 0600
From: Brooke Shipley
Title:Algebras and modules in monoidal model categories
Authors: Stefan Schwede, Brooke E. Shipley
Email: schwede@math.mit.edu
Email2: bshipley@math.uchicago.edu
We construct model category structures for monoids and modules in
symmetric monoidal model categories which satisfy an extra axiom, the monoidal
axiom. This paper was inspired in particular to deal with two of the new
symmetric monoidal categories of spectra, symmetric spectra and $\Gamma$spaces.
This paper is available at the homotopy theory archive at
http://hopf.math.purdue.edu
or via anonymous ftp at hopf.math.purdue.edu. It will also be available
through math.AT at xxx.lanl.gov.
Date: Thu, 22 Jan 1998 16:40:03 0400 (AST)
Subject: coinduction papers
Date: Wed, 21 Jan 1998 21:22:55 +0000 (GMT)
From: Dusko Pavlovic
Dear Categories,
Two papers about coinduction and guarded induction, one of them
joint work with Martin Escardo, are available from
http://www.cogs.susx.ac.uk/users/duskop/ or
ftp://ftp.cogs.susx.ac.uk/pub/users/duskop/
*Calculus in coinductive form* is not one of those funny calculi where
you can prove anything, just your old NewtonLeibnizTaylorLaplace
calculus, with a special emphasis on Taylor and Laplace, and a bit of
categories.
*Guarded induction*, on the other hand, is this logical principle
whereby, to prove a proposition p, you are allowed to use, among other
things, that very same proposition p  erm, provided, of course,
that you make sure that it is #guarded#. This gives rise to various
funny calculi, and a bit of categories.
(Proper abstracts follow.)
With best wishes,
 Dusko Pavlovic
===================================================================
CALCULUS IN COINDUCTIVE FORM
by D. Pavlovic and M.H. Escardo
Abstract.
Coinduction is often seen as a way of implementing infinite objects.
Since real numbers are typical infinite objects, it may not come as a
surprise that calculus, when presented in a suitable way, is permeated
by coinductive reasoning. What *is* surprising is that mathematical
techniques, recently developed in the context of computer science,
seem to be shedding a new light on some basic methods of calculus.
We introduce a coinductive formalization of elementary calculus that
can be used as a tool for symbolic computation, and geared towards
computer algebra and theorem proving. So far, we have covered parts
of ordinary differential and difference equations, Taylor series,
Laplace transform and the basics of the operator calculus.
===================================================================
GUARDED INDUCTION ON FINAL COALGEBRAS
by D. Pavlovic
Abstract.
We make an initial step towards categorical semantics of guarded
induction. While ordinary induction is usually modelled in terms of
least fixpoints and initial algebras, guarded induction is based on
*unique* fixpoints of certain operations, called guarded, on *final*
coalgebras. So far, such operations were treated syntactically. We
analyse them categorically. Guarded induction appears as couched in
coinduction.
The applications of the presented categorical analysis span across the
gamut of the applications of coinduction, from modelling of
computation to solving differential equations. A subsequent paper will
provide an account of some domain theoretical aspects, which are
presently left implicit.
Date: Sun, 8 Feb 1998 12:38:37 0400 (AST)
Subject: Categorification
Date: Sat, 7 Feb 1998 16:00:14 0800 (PST)
From: john baez
Here is the abstract of a paper that is now available at my website.

Categorification
John C. Baez and James Dolan
To appear in Proceedings of the Workshop on Higher Category Theory
and Mathematical Physics at Northwestern University, Evanston, Illinois,
March 1997, eds. Ezra Getzler and Mikhail Kapranov.
Categorification is the process of finding categorytheoretic analogs of
settheoretic concepts by replacing sets with categories, functions with
functors, and equations between functions by natural isomorphisms
between functors, which in turn should satisfy certain equations of
their own, called `coherence laws'. Iterating this process requires a
theory of `ncategories', algebraic structures having objects,
morphisms between objects, 2morphisms between morphisms and so on up to
nmorphisms. After a brief introduction to ncategories and their
relation to homotopy theory, we discuss algebraic structures that can be
seen as iterated categorifications of the natural numbers and integers.
These include tangle ncategories, cobordism ncategories, and the homotopy
ntypes of the loop spaces Omega^k S^k. We conclude by describing a
definition of weak ncategories based on the theory of operads.

The paper is available in Postscript form on the web at
http://math.ucr.edu/home/baez/cat.ps
I can also email or snailmail you a copy at your request.
Date: Tue, 24 Feb 1998 14:18:32 GMT
From: Samin Ishtiaq
Subject: categories: new paper: A Relevant Analysis of Natural Deduction
We apologize for multiple copies of this mail.
The following paper will appear in the Journal of Logic and
Computation (expected in Vol. 8) later this year:
A Relevant Analysis of Natural Deduction
S Ishtiaq and DJ Pym
Queen Mary and Westfield College
University of London
{si,pym}@dcs.qmw.ac.uk
We study a framework, RLF, for defining natural deduction
presentations of linear and other relevant logics. RLF consists in a
language together, in a manner similar to that of LF, with a
representation mechanism. The language of RLF, the
$\lambda\Lambda_{\kappa}$calculus, is a system of firstorder linear
dependent function types which uses a function $\kappa$ to describe
the degree of sharing of variables between functions and their
arguments. The representation mechanism is judgementsastypes,
developed for linear and other relevant logics. The
$\lambdal\Lambda_{\kappa}$calculus is a conservative extension of the
$\lambda\Pi$calculus and RLF is a conservative extension of LF.
The paper will be available from our Hypatia entries, at
http://hypatia.dcs.qmw.ac.uk. It is also available at
http://www.dcs.qmw.ac.uk/~si.
We are currently engaged in further study of the proof theory of the
$\lambda\Lambda_{\kappa}$calculus; this includes setting up a
propositionastypes correspondence and a Gentzenization of the type
theory. We are also investigating categorical models, specifically
resourcedindexed Kripke models, of the
$\lambda\Lambda_{\kappa}$calculus.
Samin Ishtiaq
David Pym
Date: Fri, 27 Feb 1998 09:33:00 0500
From: rblute@mathstat.uottawa.ca (Richard Blute)
Subject: categories: Paper available
The following paper is available by anonymous ftp at triples.math.mcgill.ca
in the directory pub/blute as nuclear.ps.gz. It is also on Prakash Panangaden's
homepage at wwwacaps.cs.mcgill.ca. Feel free to contact me if there
are any problems.
Cheers,
Rick Blute
Nuclear and Trace Ideals in Tensored *Categories
=================================================
Samson Abramsky Richard Blute
Department of Computer Science Department of Mathematics
University of Edinburgh and Statistics
Edinburgh, Scotland University of Ottawa
Ottawa, Ontario, Canada
Prakash Panangaden
Department of Computer Science
McGill University
Montreal, Quebec, Canada
Presented to Mike Barr on the occasion of his 60th birthday.
Abstract
========
We generalize the notion of nuclear maps from functional analysis by
defining nuclear ideals in tensored *categories. The motivation for
this study came from attempts to generalize the structure of the category
of relations to handle what might be called ``probabilistic relations''.
The compact closed structure associated with the category of relations
does not generalize directly, instead one obtains nuclear ideals.
Most tensored *categories have a large class of morphisms
which behave as if they were part of a compact closed category, i.e. they
allow one to transfer variables between the domain and the codomain. We
introduce the notion of nuclear ideals to analyze these classes of
morphisms. In compact closed categories, we see that all morphisms
are nuclear, and in the category of Hilbert spaces, the nuclear morphisms
are the HilbertSchmidt maps.
We also introduce two new examples of tensored *categories, in which
integration plays the role of composition. In the first, morphisms are a
special class of distributions, which we call tame distributions.
We also introduce a category of probabilistic relations which was
the original motivating example.
Finally, we extend the recent work of Joyal, Street and Verity
on traced monoidal categories to this setting by introducing the notion
of a trace ideal. For a given symmetric monoidal category, it is not
generally the case that arbitrary endomorphisms can be assigned a trace.
However, we can find ideals in the category on which a trace can be
defined satisfying equations analogous to those of Joyal, Street and
Verity. We establish a close correspondence between nuclear ideals and
trace ideals in a tensored *category, suggested by the correspondence
between HilbertSchmidt operators and trace operators on a Hilbert space.
When we apply our notion of trace ideal to the category of Hilbert spaces,
we obtain the usual trace of an endomorphism in the trace class.
Date: Sat, 28 Feb 1998 11:41:14 0500 (EST)
From: Robert Seely
Subject: categories: Paper on Feedback announced
The following paper is available on RAG Seely's WWW home page at
or directly by ftp at
or
Comments are most welcome; please send them to any of the authors.
Any problems in obtaining the paper should be sent to rags@math.mcgill.ca.
Feedback for linearly distributive categories:
traces and fixpoints
by
R.F. Blute
J.R.B. Cockett
R.A.G. Seely
ABSTRACT
In the present paper, we develop the notion of a trace operator
on a linearly distributive category, which amounts to essentially
working within a subcategory (the "core") which has the same sort of
"type degeneracy" as a compact closed category. We also explore the
possibility that an object may have several trace structures,
introducing a notion of compatibility in this case. We show that
if we restrict to compatible classes of trace operators, an object may
have at most one trace structure (for a given tensor structure). We give
a linearly distributive version of the "geometry of interaction"
construction, and verify that we obtain a linearly distributive category
in which traces become canonical. We explore the relationship between
our notions of trace and fixpoint operators, and show that an object
admits a fixpoint combinator precisely when it admits a trace and is
a cocommutative comonoid. This generalises an observation of Hyland and
Hasegawa.
This paper is presented to Bill Lawvere on the occasion of his 60th
birthday.
===================================
RAG Seely
[ NB  please use the "generic" email address above and not
machine specific eaddresses like "rags@triples.math.mcgill.ca" ]
===================================
Date: Sat, 25 Apr 1998 16:03:09 +0300 (IDT)
From: ZIPPIE Gonczarowski
Subject: categories: Preprints available
The following preprints are available. Please email to:
zippie@actcom.co.il
1. Introducing the Mathematical Category of Artificial Perceptions
by Z. ArziGonczarowski and D. Lehmann
To be published this summer in `The Annals of Mathematics and Artificial
Intelligence'.
2. From Environments to Representations  A Mathematical Theory of Artificial
Perceptions
by Z. ArziGonczarowski and D. Lehmann
To be published in `Artificial Intelligence'.
(To those of you who have already asked for them  they are being posted today).
__________________________________________________________________________
Dr. Zippora ArziGonczarowski
Typographics, Ltd.
46 Hehalutz St.
Jerusalem 96222
Israel
Tel: (+972)26437819 Fax: (+972)26434252 Email: zippie@actcom.co.il
__________________________________________________________________________
From: Giuseppe Longo
Date: Tue, 28 Apr 98 15:46:15 +0200
Subject: categories: Book available by ftp
The book below is currently out of print. Upon kind permission of
the M.I.T. Press, it is now available by ftp, via my web page (see
the book content page in Downloadable Papers).
Andrea Asperti and Giuseppe Longo. Categories, Types and
Structures: an introduction to Category Theory for the working
computer scientist. M.I.T. Press, 1991. (pp. 1300).
Giuseppe Longo
http://www.dmi.ens.fr/users/longo
email: longo@dmi.ens.fr
Date: Wed, 6 May 1998 18:15:06 0400 (EDT)
From: Steve Awodey
Subject: categories: preprint available
Dear Colleagues,
The preprint mentioned below is available from my page on the WWW,
http://www.andrew.cmu.edu/user/awodey/
Please let me know if you have difficulty obtaing or printing it, or if you
would like to have a paper copy sent.
Steve A.
*******************************************************************************
"Topological representation of the lambdacalculus"
S. Awodey
Abstract: The lambdacalculus can be represented topologically by
assigning certain spaces to the types and certain continuous maps to the
terms. Using a recent result from topos theory, the usual calculus of
lambdaconversion is shown to be deductively complete with respect to such
topological
semantics. It is also shown to be functionally complete, in the sense
that there is always a ``minimal'' topological model, in which every
continuous function is lambdadefinable. These results subsume earlier
ones using cartesian closed categories, as well as those employing
socalled Henkin and Kripke lambdamodels.
*******************************************************************************
Date: Thu, 7 May 1998 15:25:22 0400 (EDT)
From: Steve Awodey
Subject: categories: more precisely ...
Dear Colleagues,
Since this is the categories list, I could and should have been more
specific about the contents of the preprint I announced yesterday:
"Topological representation of the lambdacalculus", available from my page
on the WWW,
http://www.andrew.cmu.edu/user/awodey/
In a nutshell, the point is that the ButzMoerdijk spatial covering
theorem for topoi can be used to embed any CCC fully and faithfully (and
CC) into a topos of sheaves on a space. So the "topological semantics"
mentioned are actually in such topoi of sheaves.
Steve A.
Subject: categories: Higherdimensional paper available
Date: Fri, 5 Jun 1998 15:48:32 +0100 (BST)
From: Tom Leinster
Higherdimensional paper available, from
http://www.dpmms.cam.ac.uk/~leinster.
Structures in HigherDimensional Category Theory
This is an exposition of some of the constructions which have arisen in
higherdimensional category theory. We start with a review of the general
theory of operads and multicategories. Using this we give an account of
Batanin's definition of ncategory; we also give an informal definition in
pictures. Next we discuss Graycategories and their place in coherence
problems. Finally, we present various constructions relevant to the opetopic
definitions of ncategory.
New material includes a suggestion for a definition of lax cubical
ncategory; a characterization of small Graycategories as the small
substructures of 2Cat; a conjecture on coherence theorems in higher
dimensions; a construction of the category of trees and, more generally, of
npasting diagrams; and an analogue of the BaezDolan slicing process in the
general theory of operads.
(A few corrections have been made to the version of this distributed at the
PSSL in Utrecht, and these are listed at the web site.)
Tom Leinster
From: "JONATHON FUNK"
Date: Wed, 10 Jun 1998 17:39:14 EET +0200 DST
Subject: categories: preprint available
Dear Colleagues,
A preprint, whose abstract follows, is available in compressed .dvi
form (for DOS and for UNIX) from:
http://www.emu.edu.tr/academic/facartsc/mathsdep/staffpic/jfunk.htm
or if you are browsing the web, click on academics, teaching
staff, Mathematics, Jonathon Funk, additional information,
after you have reached the EMU homepage http://www.emu.edu.tr
If you would like a copy, but are unable to retrieve the preprint,
please don't hestitate to contact me, as I would be happy to send
you the .dvi file personally.
funk@mozart.emu.edu.tr

``On branched covers in topos theory''
Abstract: We present some new findings conerning branched covers in
topos theory. Our discussion involves a particular subtopos of a
given topos that can be described as the smallest subtopos closed
under small coproducts in the including topos. We also have some new
results concerning the general theory of KZdoctrines, such as the
the closure under composition of discrete fibrations for a KZ
doctrine (in the sense of Bunge/Funk, ``On a bicomma object condition
for KZdoctrines'').
Regards,
Jonathon Funk
Jonathon Funk
Department of Mathematics
Eastern Mediterranean University
Gazimagusa
Turkish Republic of North Cyprus
via Mersin 10, Turkey
tel: (90) 392 366 6588, Ext: 1227, 1228, 1138
fax: (90) 392 366 1604
Date: Mon, 15 Jun 1998 11:23:12 +0200 (MET DST)
From: "I. Moerdijk"
Subject: categories: new preprint
Dear categorists,
The following preprint is available from the
KTheory archive at http://www.math.uiuc.edu/Ktheory,
and hopefully from our homepages before long. Please
let us know in case you wish to be sent a hard copy.
Ieke Moerdijk.

A Homology Theory for Etale Groupoids
by Marius Crainic and Ieke Moerdijk
In this paper we introduce a homology theory for etale groupoids, dual to
Haefliger's cohomology theory (via Poincare duality). We prove basic facts
like Morita invariance, Leray spectral sequence, Verdier duality. We also
outline the application to the computation of cyclic homology of the
convolution algebra of the groupoid (including the nonHausdorff situation).
An appendix about "compact supports" on nonHausdorff manifolds is added.
Marius Crainic
Ieke Moerdijk

Date: Wed, 24 Jun 1998 10:49:40 0400 (EDT)
From: Susan Niefield
Subject: categories: preprint available
The following reprint is available at
http://www1.union.edu/~niefiels/ESU.ps
http://www1.union.edu/~niefiels/ESU.dvi
EXPONENTIABILITY AND SINGLE UNIVERSES
by Marta BUNGE and Susan NIEFIELD
ABSTRACT  The search for suitable single universes for opposite or dual
pairs of notions (such as those of discrete fibration and discrete
opfibration, or of open and closed inclusions, or of functions and
distributions on a Grothendieck topos) leads naturally to
exponentiability. Using exponentiability techniques, such as
modelgenerated categories and glueing, we settle a standing conjecture
and an open problem. The conjecture, due to F. Lamarche, states that for
a small category B, the category of unique factorization liftings (also
known as discrete Conduche fibrations) over B is a topos. We also
construct the smallest topos containing the local homeomorphisms
(functions) and the complete spreads (distributions) over any given topos
satisfying a certain condition (true of presheaf toposes). This solves a
problem posed by F. W. Lawvere. Along the way, we introduce two new sorts
of geometric morphisms, characterize locally closed inclusions in Cat,
and investigate new features of generalized coverings in topos theory,
such as branched coverings, cuts, and complete spreads.
Date: Thu, 2 Jul 1998 22:10:56 0400 (EDT)
From: Susan Niefield
Subject: categories: withdrawal of preprint
The paper "Exponentiablity and Single Universes" by Marta Bunge and Susan
Niefield, recently announced on the site ww1.union.edu/~niefiels has been
temporarily withdrawn. A revised version will be posted soon. The paper
contained an erroneous result  namely, that for an arbitrary small
category B, the category UFL/B of GiraudConduche fibrations over B is a
topos. A counterexample has been found by Peter Johnstone.
Date: Mon, 6 Jul 1998 16:49:05 +1000
From: mbatanin@mpce.mq.edu.au (Michael Batanin)
Subject: categories: generalized computads
Dear collegues,
the following preprint
"Computads for finitary monads on globular sets"
is available at
http://wwwmath.mpce.mq.edu.au/~mbatanin/papers.html
>From Introduction.
This work arose as a reflection on the foundation of higher
dimensional category theory. One of the main ingredients of any
proposed definition of weak $n$category is the shape of diagrams
(pasting scheme) we
accept to be composable. In a globular approach \cite{Bat} each
$k$cell
has a source and target $(k1)$cell. In the opetopic approach of
Baez and Dolan \cite{BD} and the multitopic approach of Hermida,
Makkai
and Power \cite{HMP} each $k$cell has a unique $(k1)$cell as
target
and a whole $(k1)$dimensional pasting diagram as source.
In the theory of strict $n$categories both source and target may be a
general pasting diagram \cite{J,StH, StP}.
The globular approach
being the simplest one
seems too restrictive to describe the combinatorics of higher
dimensional compositions. Yet, we argue that this is a false
impression. Moreover, we prove that this approach is a basic
one from which the other type of composable diagrams may be derived.
One theorem proved here asserts that the category of algebras of
a finitary monad on the category of $n$globular sets is {\bf
equivalent} to the category of algebras of an appropriate monad on
the
special category (of computads) constructed from the data of the
original monad. In the case of the monad derived from the universal
contractible operad \cite{Bat} this result may be interpreted as the
equivalence of the definitions of weak $n$categories (in the sense
of \cite{Bat}) based on the
`globular' and general pasting diagrams. It may be also considered
as the first step toward the proof of equivalence of the different
definitions of weak $n$category.
We also develop a general theory of computads and investigate some
properties of the category of generalized computads. It turned out,
that in a good situation this category is a topos (and even a presheaf
topos under some not very restrictive conditions, the property firstly
observed by S.Schanuel and reproved by A,Carboni and P.Johnstone for
$2$computads in the sense of Street).
/\
/ \
M / Co \> MQ
/ A C T\
/________\
Centre of Australian Category Theory
Mathematics Department, Macquarie University
New South Wales 2109, AUSTRALIA
Date: Tue, 21 Jul 1998 13:55:47 0400 (EDT)
From: Susan Niefield
Subject: categories: revised preprint available
A revised version of the following reprint is available at
http://www1.union.edu/~niefiels/ESU.ps
http://www1.union.edu/~niefiels/ESU.dvi
EXPONENTIABILITY AND SINGLE UNIVERSES
by Marta BUNGE and Susan NIEFIELD
ABSTRACT  In this paper, we first consider known universes for pairs of
opposite notions such as those of discrete fibrations/discrete
opfibrations and of open/closed locale inclusions, and then extrapolate
these in order to introduce new single universes for open/closed
inclusions of subcategories and for functions/distributions on a topos. A
key factor that these notions have in common is exponentiability in the
ambient category. Along the way, we (1) prove that, for a factorization
linearly ordered small category B, the category of discrete
GiraudConduche fibrations over B is a (model generated) topos, (2)
characterize locally closed inclusions in the category Cat of small
categories, (3) investigate ``generalized coverings'' in topos theory,
including branched coverings, cuts, and complete spreads, and (4) examine
the preservations of exponetiability under the passage from Cat/B to the
category of Grothendieck toposes over the presheaves PB.
Date: Mon, 03 Aug 1998 14:18:11 +0100
From: "David J. Pym"
Subject: categories: The Logic of Bunched Implications.
We, Peter O'Hearn and David Pym, are pleased to announce
our new paper, ``The Logic of Bunched Implications''.
We hope it will be of interest to readers of `categories'. BI
is a relevant logic which extends both linear and intuitionistic
logic. It has a semantics of proofs based on `doubly closed categories',
which carry two monoidal closed structures, one of which is cartesian
in models of BI. A rich class of models is provided by Day's tensor
product construction on the category of presheaves over a small
monoidal category. It also comes with a lambda calculus, a truth
semantics
and a computational interpretation as a logic of resources, quite
different from
that of linear logic.
The paper is available at:
http://www.dcs.qmw.ac.uk/~pym
and
http://www.dcs.qmw.ac.uk/~ohearn
where drafts of various companion and related papers can/will be
found.
P.W.O'Hearn and D.J. Pym
Queen Mary & Wesfield College,
University of London
Abstract.
Introduce a logic BI in which a multiplicative (or linear)
and an additive (or intuitionistic) implication live side by
side. The propositional version of BI arises from an analysis
of the prooftheoretic relationship between conjunction and
implication, and may be viewed as a merging of intuitionistic
logic and multiplicative, intuitionistic linear logic. The
predicate version of BI includes, in addition to standard
additive quantifiers, multiplicative (or intensional)
quantifiers ``forallnew'' and ``existnew'' which arise from
observing restrictions on structural rules on the level of terms
as well as propositions. We discuss computational interpretations,
based on sharing, at both the propositional and predicate levels.
Date: Mon, 3 Aug 1998 19:01:17 +0100 (BST)
From: Ronnie Brown
Subject: categories: New Preprint
New preprint
R. Brown and I. Icen.
"Lie local subgroupoids and their monodromy",
UWB Math Preprint 98.15, 12pp.
ABSTRACT:The notion of local equivalence relation on a topological
space is generalised to that of local
subgroupoid. Properties of coherence are considered. The main result is
notions of holonomy and monodromy groupoid for certain Lie local
subgroupoids.
http://www.bangor.ac.uk/~mas010/papers/sub6.ps,.dvi
Prof R. Brown, School of Mathematics,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474office: 382475
fax: +44 1248 383663
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm
Date: Tue, 4 Aug 1998 18:01:18 +0100
Subject: categories: Injectives via KZmonads
From: "Martin Escardo"
The following short paper is available from my home page:
http://www.dcs.ed.ac.uk/home/mhe/pub/papers/top97.ps.gz
or
http://www.dcs.ed.ac.uk/home/mhe/papers.html
(It is an updated version of a paper which was previously circulated
in other lists.)
=====================================
Injective spaces via the filter monad
======================================================================
An injective space is a topological space with a strong extension
property for continuous maps with values on it. A certain filter space
construction embeds every T_0 topological space into an injective
space. The construction gives rise to a monad. We show that the monad
is of the KockZoberlein type and apply this to obtain a simple proof
of the fact that the algebras are the continuous lattices (Alan Day,
1975, Oswald Wyler, 1976). In previous work we established an
injectivity theorem for monads of this type, which characterizes the
injective objects over a certain class of embeddings as the
algebras. For the filter monad, the class turns out to consist
precisely of the subspace embeddings. We thus obtain as a corollary
that the injective spaces over subspace embeddings are the continuous
lattices endowed with the Scott topology (Dana Scott, 1972). Similar
results are obtained for continuous Scott domains, which are
characterized as the injective spaces over dense subspace embeddings,
via the proper filter monad.
======================================================================
Two notes (and some questions concerning credit)
=========
(i) Bob Flagg and I have also considered the following variations on
the filter monad (a report is being written)
(a) Category: T_0 exponentiable spaces
(= corecompact = open sets form a continuous lattice)
Restriction on filters: Scott open.
=> Associated maps: "semiproper" embeddings
(= right adjoint of the frame maps preserve directed joins)
=> Algebras (and hence injectives over semiproper):
continuous meetsemilattices with Scott topology.
(Corollary: continuous meetsemilattices and Scott continuous
functions form a CCC. Was this known before?)
This characterization of the algebras was previously known
(Andrea Shalkanyone else?), but the "KZmethod" outlined
in the above abstract gives a much shorter proof.
(b) Category: T_0 spaces
Restriction on filters: prime.
=> Associated maps: flat embeddings
(= right adjoint of the frame maps preserve finite joins)
=> Algebras (and hence injectives over flat):
compact, stably locally compact spaces.
(A localic version is given via the ideal monad. What Johnstone
refers to as Joyal's Lemma appears as a special case of this.)
(I don't know what was previously known about this.)
(A result by Isbell (in his paper "Flat = prosupersplit")
implies that the flat embeddings form the largest class of
embeddings over which the CSLCSs are injective, because
finite spaces are (trivially) CSLCSs.)
(c) Category: T_0 spaces
Restriction on filters: completely prime.
=> Associated maps: "completely flat" embeddings
(= right adjoint of the frame maps preserve all joins)
=> Algebras (and hence injectives over completely flat):
sober spaces.
(d) Category: T_0 locally connected spaces
Restriction on filters: filters of connected open sets.
=> Associated maps: "locally dense" embeddings
(= frame maps preserve connectedness (and hence right
adjoints preserve disjoint unions))
=> Algebras (and hence injectives over locally dense):
Ldomains.
(This was obtained by Bob, based on some previous work by
Paul Taylor (and Andrea Shalk) on the algebras. Again, the
KZmethod gives a simpler proof of the characterization.)
(ii) The filter monad is formally analogous to the socalled
continuation monad, as it is observed (with the formal details of the
analogy) in the paper being advertised.
I would like to also mention that the general injectivity result for
KZmonads referred to in the above abstract was established in the
paper
http://www.dcs.ed.ac.uk/home/mhe/pub/papers/injective.ps.gz
which is (mainly) about continuity of the extension process (answering
a question by Scott in his 1972 paper on continuous lattices).
Comments are wellcome.
Martin
=================================================================
Martin H. Escardo, Department of Computer Science, LFCS
King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
office: 2606 (JMCB) fax: +44 131 667 7209 phone: +44 131 650 5135
mailto:mhe@dcs.ed.ac.uk http://www.dcs.ed.ac.uk/home/mhe
=================================================================
Date: Fri, 7 Aug 1998 09:44:50 +0100 (BST)
From: Ronnie Brown
Subject: categories: New Preprint (revised)
The url for the following was not correct and is revised below:
New preprint
R. Brown and I. Icen.
"Lie local subgroupoids and their monodromy",
UWB Math Preprint 98.15, 12pp.
ABSTRACT:The notion of local equivalence relation on a topological
space is generalised to that of local
subgroupoid. Properties of coherence are considered. The main result is
notions of holonomy and monodromy groupoid for certain Lie local
subgroupoids.
http://www.bangor.ac.uk/~mas010/brownpr.html#monodromy
Prof R. Brown, School of Mathematics,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474office: 382475
fax: +44 1248 383663
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm
Date: Wed, 28 Oct 1998 15:10:20 0500 (EST)
From: Michael Barr
Subject: categories: Papers available
I have just posted two new papers on ftp.math.mcgill.ca. The first is
called balls.dvi and is my joint paper with Heinrich Kleisli on
*autonomous categories of topological balls. It will appear, bye and bye,
in the Cahiers. The second, called chu_se.dvi is on the separated
extension Chu category and is to be published in TAC within the next day
or two.
Michael
Date: Fri, 30 Oct 1998 08:24:52 0400
From: Marta Bunge
Subject: categories: Paper available
This is to announce a new paper, by
Marta Bunge and Marcelo Fiore, "Unique factorization Lifting Functors
and Categories of Processes".
http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.dvi
http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.ps
The paper is organised as follows. After an Introduction, Section 1
presents background material motivated from the point of view of computer
science. In Section 2, the category UFL of unique factorisation lifting
(ufl) functors is recalled and its basic properties are studied. Section 3
explores applications of ufl functors to concurrency. In particular we
show that they may be used in the study of interleaving models like
transition systems. In Section 4, we introduce triangulated categories.
Our main use for them is in Section 5 where, for C a triangulated category,
we exhibit the category UFL/C as a sheaf topos. These toposes may be
regarded as models of linearlycontrolled processes. Some
concluding remarks are provided in Section 6.
Professor Marta Bunge
McGill University
Department of Mathematics & Statistics
Burnside Hall
805 Sherbrooke Street West
Montreal, QC
Canada H3A 2K6
Fax: (514) 933 8741
Phone: (514) 933 6191
Date: Thu, 12 Nov 1998 15:26:28 +0100
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: Preprint available
The following preprint:
M. Grandis,
"An intrinsic homotopy theory for simplicial complexes
with applications to image processing"
is available at:
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/
as: Lnk.Nov98.ps
***
Abstract. A simplicial complex is a set equipped with a downclosed family
of distinguished finite subsets; this structure is mostly viewed as
codifying a triangulated space. Here, this structure is used directly to
describe "spaces" of interest in various applications, where the associated
triangulated space would be misleading. An intrinsic homotopy theory, not
based on topological realisation, is introduced.
The applications considered here are aimed at metric spaces and
digital topology; concretely, at image processing and computer graphics. A
metric space X has a structure t_e(X) of simplicial complex at each
"resolution" e > 0; the resulting nhomotopy group \pi_n(t_e(X)) detects
those singularities which can be captured by an ndimensional grid, with
edges bound by e; this works equally well for continuous or discrete
regions of euclidean spaces.
***
Comments would be appreciated.
In particular, I am uneasy about a question of terminology.
In my opinion, the term "simplicial complex", quite appropriate when the
structure is viewed as codifying a triangulated space, is unfit when such
objects are treated as "spaces" in themselves (somewhat close to
bornological spaces, which have similar axioms on objects and maps).
In other words, "simplicial complex" should not refer to the category
itself, say C, but to its usual embedding in Top, the simplicial
realisation. The two aspects may clash, e.g. with respect to initial or
final structures: the coarse Cobject on three points (final structure, all
parts are distinguished) is realised as a euclidean triangle; a Csubobject
is sufficient to produce a topological subspace (a regular subobject in
Top), but a Csubspace (a regular subobject in C) is a stronger notion.
Moreover, from a more concrete point of view, the simplicial realisation is
quite inappropriate in most of the applications considered in this work.
The opposition "Cobject / simplicial complex" is in part similar to
"sequence / series": the second term refers to a more specific view & use
of the same data; the clashing of the opposition is particularly evident in
the notions of convergence, for a sequence or a series.
That's why I am calling a Cobject a "combinatorial space". (The term
"combinatorial complex" has also been used for simplicial complex; and I
wanted a term of the form "attribute + space", to use freely of topological
terms like discrete, coarse, subspace...)
But of course it is embarassing to propose a new term for a classical notion.
Marco Grandis
Date: Tue, 17 Nov 1998 09:26:51 +0100
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: Re: preprint available (on simplicial complexes)
Reply to James Stasheff
> is it also available on your web page without ftp?
No, from my home page:
http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/
you would just have access to all papers available by ftp, at the
address I gave:
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/
***
> will you be posting it to the math archive at lanl?
yes, in a while and possibly after revision.
***
With best wishes
Marco Grandis
Date: Wed, 18 Nov 1998 13:27:09 +0100
From: kock
Subject: categories: preprint available (Kock and Reyes)
The following preprint
A. Kock and G.E. Reyes: Fractional Exponent Functors and Categories of
Differential Equations
is available via the Home Page
http://www.mi.aau.dk/~kock/
or directly by ftp (200 KB)
ftp://ftp.imf.au.dk/pub/kock/ODE5.ps
(also available in .dvi format, 100 KB).
Abstract: This paper grew out of a question/suggestion of Lawvere: to use
the "amazing right adjoints" (= fractional exponents) of Synthetic
Differential Geometry, to get information on the category of second order
differential equations. As a byproduct of our investigations, we derive
some information about the strength (enrichment) of fractional exponent
functors in general.
Date: Wed, 18 Nov 1998 16:09:09 +0000 (GMT)
Subject: categories: preprint available
From: "Martin Escardo"
The following preprint is available at
http://www.dcs.ed.ac.uk/home/mhe/pub/papers/patchCSLC.ps.gz
& http://www.dcs.ed.ac.uk/home/mhe/papers.html
& ftp://ftp.dcs.ed.ac.uk/pub/mhe/patchCSLC.ps.gz
On the compactregular coreflection
of a compact stably locally compact locale.
ABSTRACT: The Scott continuous nuclei form a subframe of the frame of
all nuclei. We refer to this subframe as the patch frame. We show that
the patch construction exhibits (i) the category of Stone locales and
continuous maps as a coreflective subcategory of the category of
coherent locales and coherent maps, (ii) the category of compact
regular locales and continuous maps as a coreflective subcategory of
the category of compact stably locaaly compact locales and perfect
maps, and (iii) the category of regular locally compact locales and
continuous maps as a coreflective subcategory of the category of
stably locally compact locales. We relate our patch construction to
Banaschewski and Brümmer's construction of the dual equivalence of the
category of compact stably locally compact locales and perfect maps
with the category of compact regular biframes and biframe
homomorphisms.
Comments are welcome.

Martin H. Escardo, LFCS, Computer Science, Edinburgh University
King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
office: 2606 (JMCB) fax: +44 131 667 7209 phone: +44 131 650 5135
mailto:mhe@dcs.ed.ac.uk http://www.dcs.ed.ac.uk/home/mhe

Date: Thu, 19 Nov 1998 22:49:16 0500 (EST)
From: F W Lawvere
Subject: categories: announcement of preprints
There are two items now available for downloading (PDF) from
my homepage
http://www.acsu.buffalo.edu/~wlawvere/
They are:
Volterra's functionals and the covariant cohesion of space
Abstract:
Volterra's principle of passage from finiteness to infinity is far
less limited than a linearized construal of it might suggest; I outline
in Section III a nonlinear version of the principle with the help of
category theory. As necessary background I review in Section II some of
the mathematical developments of the period 18871913 in order to clarify
some more recent advances and controversies which I discuss in Section I.
Some relevant historical and current literature is discussed in relation
to the categorical analysis: Volterra and Hadamard on the notion of
element, Fichera's critique of the relation between functional analysis
and continuum physics, and the recent Michor & Kriegl book published by
the AMS.
Outline of Synthetic Differential Geometry
Abstract:
These rough notes were distributed to the geometry seminar at Buffalo
in February 1998, sketching the background of categorical dynamics in
anticipation of the April 1999 AMS Meeting in which there will be a
special session on related matters. In particular, some of the results
stated in my September 1997 AMS talk in Montreal "Toposes of laws of
motion" are outlined, especially the relation of second order differential
equations to a.t.o.m's (amazingly tiny object models). An additional
appendix has been added (November 1998) to these rough notes concerning
recent advances on these questions by Kock and Reyes.
Toposes of laws of motion
will be added to the web page as soon as the transcript of the original
video is completed.
**********************************************************************
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
7168292144 ext. 117 Buffalo, N.Y. 14214, USA
**********************************************************************
Date: Fri, 20 Nov 1998 15:53:55 0500 (EST)
From: Michael Barr
Subject: categories: staragain.dvi
A paper titled
*Autonomous categories: once more around the track has just been posted:
ftp.math.mcgill.ca/pub/barr/staragain.dvi
Basically, it redoes nearly all of the original Lecture Notes volume in
about one fifth the space, using the Chu construction and proving a very
general theorem. The only example from the original monograph that is not
convered by this theorem is the category of Banach balls, the subject of a
recent paper by Kleisli and me. That is also thee under the name
balls.dvi (if I have remembered it correctly).
This paper is to be submitted to tac for the Lambekfestschrift.
From: john baez
Subject: categories: HDA4: 2Tangles
Date: Mon, 23 Nov 1998 20:08:31 0800 (PST)
The following preprint is now available at the places listed below.
Comments and corrections are welcome!

HigherDimensional Algebra IV: 2Tangles
John C. Baez, Laurel Langford
Just as knots and links can be algebraically described as certain
morphisms in the category of tangles in 3 dimensions, compact surfaces
smoothly embedded in R^4 can be described as certain 2morphisms in
the 2category of `2tangles in 4 dimensions'. Using the work of
Carter, Rieger and Saito, we prove that this 2category is the `free
semistrict braided monoidal 2category with duals on one unframed
selfdual object'. By this universal property, any unframed selfdual
object in a braided monoidal 2category with duals determines an
invariant of 2tangles in 4 dimensions.

This paper is math.QA/981139 on the mathematics preprint server,
so you can get it at:
http://xxx.lanl.gov/abs/math.QA/9811139
It's also available as a Postscript file at my website:
http://math.ucr.edu/home/baez/hda4.ps
If you have any trouble, let me know and I can send you a copy.
Date: Wed, 9 Dec 1998 22:22:03 0800 (PST)
From: Jonathon Funk
Subject: categories: preprint (revised) available
The preprint: On Branched Covers in Topos Theory
is available from my home page
http://www.math.ubc.ca/~funk/
in postscript or .dvi format.
This is a revision and extension of a preprint I had
posted from Cyprus in June, 98.
Abstract: We present some new findings concerning branched covers in
topos theory. Our discussion involves a particular subtopos of a given
topos that can be described as the smallest subtopos closed under small
coproducts in the including topos.
Our main result is a description of the covers of this subtopos as a
category of fractions of branched covers, in the sense of R. Fox, of the
including topos. We also have some new results concerning the general
theory of KZdoctrines, such as the closure under composition of discrete
fibrations for a KZdoctrine, in the sense of BungeFunk.
Date: Mon, 14 Dec 1998 17:51:26 0500 (EST)
From: F W Lawvere
Subject: categories: Preprint available
PREPRINT AVAILABLE
A transcript of the video of my talk at the September 1997
AMS Meeting in Montreal is now available for downloading in pdf format.
The title is
TOPOSES OF LAWS OF MOTION
I will be very grateful for comments and suggestions on this paper,
as well as on the other two papers available:
http://www.acsu.buffalo.edu/~wlawvere
*******************************************************************************
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
7168292144 ext. 117 Buffalo, N.Y. 14214, USA
*******************************************************************************
Subject: categories: Preprint available (Fiore, Plotkin, and Turi).
From: Marcelo Fiore
Date: Tue, 15 Dec 1998 21:12:32 +0000
The following preprint
Abstract Syntax and Variable Binding by M.Fiore, G.Plotkin., and D.Turi.
is available as
http://www.dcs.ed.ac.uk/~dt/abstractsyn.ps
Synopsis: We show that categorical algebra in the objectclassifier topos
provides a suitable mathematical universe for modelling algebraic structures
with binding operators.
Date: Wed, 16 Dec 1998 12:10:31 +0100 (MET)
From: Jaap van Oosten
Subject: categories: paper on SDT available
The following paper is available:
Axioms and (Counter)examples in Synthetic Domain Theory
by Jaap van Oosten and Alex K. Simpson
the paper can be found at the URL:
http://www.math.uu.nl/publications/preprints/1080.ps.gz
ABSTRACT: Chapter 1 presents a development of basic Synthetic Domain
Theory on the basis of 4 axioms (1:\Sigma is complete; 2:\Sigma is
\neg\negseparated; 3:\bot\in\Sigma; 4:the Phoa Principle). New results
are, that 1 and 2 imply that the \Sigmaorder on \Sigma , I and F (the
initial lift algebra and the final lift coalgebra, respectively) is
(pointwise) implication, that 1 and 2 imply that complete extensional
objects (we call them complete regular \Sigmaposets) are stable under
lifting, that under 1,2,3, axiom 4 is equivalent to \Sigma having binary
joins, and that if \Sigma is closed under Nidexed joins in \Omega, then
all complete objects are stable under lifting. We also present an analysis
of when I is an internal colimit of a diagram 0>L(0)>L^2(0)>...
Chapters 2,3,4 investigate models. We study models of the axioms in: the
Modified realizability topos Mod, the Effective topos Eff, and a particular
Grothendieck topos. In Mod, the Scott principle fails and L(2) is not
complete. In Eff, we have that the internal colimit of 0>L(0)>L^2(0)>...
is complete (whence it is not isomorphic to I), and a general theorem
characterizing I for \neg\negseparated dominances. Finally, in a sheaf
topos we have an example where L(2) is complete but L(N) isn't.
Jaap van Oosten
Date: Fri, 18 Dec 1998 17:44:08 0800
From: Dusko Pavlovic
Subject: categories: preprint
Dear All,
As many of you know, December is the season of two column logic/CS
related preprints. The title of mine is:
Towards semantics of guarded induction
and it is at the bottom of the page
http://www.kestrel.edu/HTML/people/pavlovic/
Comments **most** welcome, esp. as I am still a bit in the darkness as
to how to present some parts. This is still an extended abstract, but a
bit more extended and less abstract than the version some of you have
seen before. (Thanks again for the questions that helped me improve it!)
With the very best wishes,
 Dusko
==============================================================================
Towards semantics of guarded induction
by Dusko Pavlovic
Abstract.
We analyze guarded induction, a coalgebraic method for implementing
abstract data types with infinite elements (e.g. various dynamic
systems, continuous or discrete). It is widely used not just in
computation, but also, tacitly, in many basic constructions of
differential calculus. However, while syntactic characterisations
abound, only the very first steps towards a formal semantics have been
made. A language independent analysis was recently initiated, but just
special cases were covered so far.
In the present paper, we propose a new approach, based on a somewhat
unusual
combination of monads and polynomial categories. The first result is
what appears to be a precise semantic characterisation of guarded
operators on arbitrary final coalgebras.
Date: Tue, 22 Dec 1998 11:36:02 +0000 (GMT)
From: Anne Heyworth
Subject: categories: thesis (involving rewriting and Kan extensions)
New PhD thesis to be found at:
http://xxx.lanl.gov/abs/math.CT/9812097
Summary of details:
Title: Applications of Rewriting Systems and Groebner Bases to Computing
Kan Extensions and Identities Among Relations.
Authors: Anne Heyworth (University of Wales, Bangor).
Comments: PhD thesis, 104 pages, LaTeX2e.
Reportno: University of Wales, Bangor preprint number 9823.
Subjclass: Category Theory; Combinatorics.
MSCclass: 1804 (Primary) 0502; 20F05; 68Q42; 68Q40; 16S15 (Secondary).
\\
This thesis concentrates on the development and application of Groebner bases
methods to a range of combinatorial problems (involving groups, semigroups,
categories, category actions, algebras and Kcategories).
Chapter Two contains the generalisation of rewriting and
KnuthBendix procedures to Kan extensions.
Chapter Three shows that the standard KnuthBendix algorithm is
stepforstep a special case of the Buchberger's algorithm for noncommutative
Groebner bases.
The onesided cases and higher dimensions are considered, and the relations
between these are made precise.
Chapter Four relates rewrite systems, Groebner bases and automata.
Reduction machines for rewrite systems are identified with standard
output
automata and the reduction machines devised for algebras are expressed as
Petrinets.
Chapter Five introduces logged rewriting for group presentations.
The completion of a logged rewriting system for a group
determines a partial contracting homotopy which enables the computation
of a set
of generators for the module of identities among relations using the
covering
groupoid methods devised by Brown and Razak Sallah.
Reducing the resulting set of submodule generators is identified as a
Groebner basis problem.

Anne Heyworth.
Subject: categories: Paper Announcement
Date: Wed, 06 Jan 1999 18:04:29 +0000
From: Alex Simpson
The following paper is available by anonymous FTP or over the Web
Lambda Definability with Sums via Grothendieck Logical Relations
by Marcelo Fiore and Alex Simpson
We introduce a notion of *Grothendieck logical relation* and use
it to characterise the definability of morphisms in *stable* bicartesian
closed categories by terms of the simplytyped lambda calculus with
finite products and finite sums. Our techniques are based on concepts
from topos theory, however our exposition is elementary.
The paper is written in a style appropriate for the conference
Typed LambdaCalculi and Applications
where it is to be presented in April. However, we briefly discuss
the true categorical content of the paper, which will be further
expanded upon in a full version of the paper (forthcoming).
The paper is available over the Web:
http://www.dcs.ed.ac.uk/~mf/TYPES/glr.{dvi,ps}
http://www.dcs.ed.ac.uk/~als/Research/glr.ps.gz
or by anonymous FTP:
ftp://ftp.dcs.ed.ac.uk/pub/mf/TYPES/glr.{dvi,ps}
ftp://ftp.dcs.ed.ac.uk/pub/als/Research/glr.ps.gz
Best wishes for a happy New Year,
Alex Simpson

Alex Simpson, LFCS, Division of Informatics, University of Edinburgh
Email: Alex.Simpson@dcs.ed.ac.uk Tel: +44 (0)131 650 5113
FTP: ftp.dcs.ed.ac.uk/pub/als Fax: +44 (0)131 667 7209
URL: http://www.dcs.ed.ac.uk/home/als
Date: Tue, 12 Jan 1999 12:32:25 0500
From: Zhaohua Luo
Subject: categories: categorical geometry
In a recent paper (TAC, Vol 4, 208248)
On Generic Separable Objects, by Robbie Gates,
the author mentioned a well known "boolean algebraic structure of the
summands of an object in an extensive category". This reminded me a
paper I posted to my homepage last year (8/30/98, see the abstract
below), in which the same boolean structure was reconstructed (at that
time this was not "well known" to me), and was applied to define the
Pierce topology for any extensive category, extending some results of
Diers. The paper
Pierce Topologies of Extensive Categories
is available at Categorical Geometry Homepage at the following new
(permanent, hopefully) address
http://www.geometry.net (or http://www.azd.com)
(The new service is a little bit slow, but offers more functions than
the old one, so please be patient.)
Best wishes,
Zhaohua Luo

Pierce Topologies of Extensive Categories
by Zhaohua Luo
Abstraction
An extensive category is a category with finite stable disjoint sums. In
this note we show that each extensive category carries a natural
subcanonical coherent Grothendieck topology defined by injections of
sums. This Grothendieck topology is induced by a strict metric topology,
which is a functor to the category of Stone spaces. We call this metric
topology the Pierce topology of the category, as it generalizes the
classical Pierce spectrums of commutative rings. Recall that the Pierce
spectrum of a commutative ring R is the spectrum of the Boolean algebra
of idempotents of R, which is a Stone space. A theorem of R. S. Pierce
states that R can be represented as the ring of global sections of a
sheaf of commutative rings on its Pierce spectrum (called the Pierce
sheaf or representation), whose stalks are indecomposable rings (with
respect to product decomposations). Diers showed that Pierce's theorem
can be extended to any object in a locally finitely presentable category
such that the opposite of the subcategory of finitely presentable
objects is lextensive (called a locally indecomposable category). We
shall see that a weak form of Pierce representation exists for any
object in an extensive category.
Subject: categories: Paper Announcement
Date: Wed, 20 Jan 1999 12:28:53 +0000
From: Alex Simpson
The following paper is available by anonymous FTP or over the Web
Computational Adequacy in an Elementary Topos
We place simple axioms on an elementary topos which suffice for it to
provide a denotational model of callbyvalue PCF with sum and product
types. The model is synthetic in the sense that types are interpreted
by their settheoretic counterparts within the topos. The main result
characterises when the model is computationally adequate with respect
to the operational semantics of the programming language. We prove that
computational adequacy holds if and only if the topos is $1$consistent
(i.e. its internal logic validates only true $\Sigma^0_1$sentences).
This paper is to appear in the forthcoming proceedings of CSL 98.
It is available from:
http://www.dcs.ed.ac.uk/~als/Research/adequacy.ps.gz
ftp://ftp.dcs.ed.ac.uk/pub/als/Research/adequacy.ps.gz
Best wishes,
Alex Simpson

Alex Simpson, LFCS, Division of Informatics, University of Edinburgh
Email: Alex.Simpson@dcs.ed.ac.uk Tel: +44 (0)131 650 5113
FTP: ftp.dcs.ed.ac.uk/pub/als Fax: +44 (0)131 667 7209
URL: http://www.dcs.ed.ac.uk/home/als
Subject: categories: Generalized Enrichment
Date: Fri, 29 Jan 1999 12:29:26 +0000 (GMT)
From: Tom Leinster
The following is now available, at http://www.dpmms.cam.ac.uk/~leinster/
Generalized Enrichment for Categories and Multicategories
In this paper we answer the question: `what kind of a structure can a general
multicategory be enriched in?' (Here `general multicategory' is used in the
sense of the author, Burroni or Hermida.) The answer is, in a sense to
be made precise, that a multicategory of one type can be enriched in a
multicategory of the type one level up. In particular, we will be able to
speak of a T_nmulticategory enriched in a T_{n+1}multicategory, where
T_n is the monad expressing the pastingtogether of nopetopes.
The answer for general multicategories reduces to something surprising in the
case of ordinary categories: a category may be enriched in an
`fcmulticategory', a very general kind of 2dimensional structure
encompassing monoidal categories, plain multicategories, bicategories and
double categories. It turns out that fcmulticategories also provide a
natural setting for the bimodules construction. We also explore enrichment
for some multicategories other than just categories. An extended application
is given: the relaxed multicategories of Borcherds and Soibelman are
explained in terms of enrichment.
Tom Leinster
PS  There's been the odd problem in the past with the web address; if it
doesn't work, try substituting "can" for "www", or send me an email.
From: Koslowski
Subject: categories: paper announcement
Date: Mon, 1 Feb 1999 01:00:07 +0100 (MET)
A heavily revised version of my paper "Beyond the Chuconstruction"
is now available from my home page:
http://www.iti.cs.tubs.de/~koslowj/RESEARCH/
It will eventually be published in Applied Categorical Structures.
I have not attempted to attribute the term "dualizing object" to anyone
in particular. The open problem of an earlier version, as to whether
Cauchycomplete bicategories of interpolads inherit local *autonomy
from their base, has been answered affirmatively.
Here is the abstract:
Starting from symmetric monoidal closed (= autonomous) categories,
PoHsiang Chu showed how to construct new *autonomous categories,
i.e., autonomous categories that are selfdual because of a
dualizing object. Recently, Michael Barr extended this to the
nonsymmetric, but closed, case, utilizing monads and modules
between them. Since these notions are wellunderstood for
bicategories, we introduce a notion of local *autonomy for these
that implies closedness and, moreover, is inherited when forming
bicategories of monads and of interpolads. Since the initial step
of Barr's construction also carries over to the bicategorical
setting, we recover his main result as an easy corollary.
Furthermore, the Chuconstruction at this level may be viewed as a
procedure for turning the endo1cells of a closed bicategory into
the objects of a new closed bicategory, and hence conceptually is
similar to constructing bicategories of monads and of interpolads.
Best regards,
 J"urgen

J"urgen Koslowski % If I don't see you no more in this world
ITI % I meet you in the next world
TU Braunschweig % and don't be late!
koslowj@iti.cs.tubs.de % Jimi Hendrix (Voodoo Child)
From: john baez
Subject: categories: higherdimensional algebra and Planckscale physics
Date: Fri, 5 Feb 1999 17:26:13 0800 (PST)
Here is a new paper of mine that may be interesting to fans of
categories and ncategories:
HigherDimensional Algebra and PlanckScale Physics
to appear in "Physics Meets Philosophy at the Planck Scale", eds.
Craig Callender and Nick Huggett, Cambridge U. Press, Cambridge.
This is a nontechnical introduction to recent work on quantum gravity
that uses ideas from higherdimensional algebra. We argue that reconciling
general relativity with the Standard Model requires a `backgroundfree
quantum theory with local degrees of freedom propagating causally'. We
describe the insights provided by work on topological quantum field theories
such as quantum gravity in 3dimensional spacetime. These are backgroundfree
quantum theories lacking local degrees of freedom, so they only display some
of the features we seek. However, they suggest a deep link between the
concepts of `space' and `state', and similarly those of `spacetime' and
`process', which we argue is to be expected in any backgroundfree quantum
theory. We sketch how higherdimensional algebra provides the mathematical
tools to make this link precise. Finally, we comment on attempts to formulate
a theory of quantum gravity in 4dimensional spacetime using `spin networks'
and `spin foams'.
Available in LaTeX and Postscript form at
http://xxx.lanl.gov/abs/grqc/9902017
and in Postscript form at
http://math.ucr.edu/home/baez/planck.ps
From: Peter Selinger
Subject: categories: Paper Announcement: Control Categories and Duality
Date: Tue, 23 Feb 1999 00:09:47 0500 (EST)
Dear Category Theorists,
The paper "Control Categories and Duality: on the Categorical
Semantics of the LambdaMu Calculus" is now available from
http://www.math.lsa.umich.edu/~selinger/papers.html
http://hypatia.dcs.qmw.ac.uk/author/SelingerP
This is a revised and improved version of a paper I presented at
MFPS'98.
Let me briefly summarize the part of the paper that I think will be
the most interesting to category theorists; the actual abstract is
appended at the end.
Consider a category C with distributive finite products and coproducts
and a distinguished object R, such that all exponentials of the form
R^A exist. Let R^C denote the full subcategory of C consisting of
objects of the form R^A. It is an old observation that R^C is
cartesianclosed. This observation is at the heart of continuation
passing style (CPS) interpretations of programming languages with
control operators, and it has been used recently by Hofmann and
Streicher to give a sound and complete categorical model of Parigot's
lambdamu calculus. (The lambdamu calculus generalizes the
simplytyped lambda calculus; it is a proof term calculus for
propositional classical logic).
In this paper, I give an independent, algebraic characterization of
the structure of categories of the form R^C. By "independent", I mean
that it does not depend on an ambient category C, and by "algebraic",
that it is given in terms of operations and equations only. The
crucial structure that R^C has, besides cartesian closure, is an
operation called "classical disjunction" that takes R^A and R^B to
R^(AxB). This operation is functorial in each argument, but not
bifunctorial; it forms a premonoidal structure in the sense of Power
and Robinson. Abstracting from R^C, I define the class of "control
categories", which are cartesianclosed categories with a premonoidal
structure and suitable axioms. The presence of an operation which is
not bifunctorial leads to some interesting twists, as one has to be a
bit careful about how one defines concepts such as weak
structurepreserving functors and equivalences of categories.
The main theorem is a structure theorem, which shows that every
control category is equivalent to a category of the form R^C (and, of
course, vice versa).
An algebraic class of categories calls out for an internal language. I
prove that the callbyname lambdamu calculus (with product and
disjunction types) forms an internal language for control
categories. Thus, these categories can be considered as models for (a
certain kind of) proof theory of classical logic. Also, the
callbyvalue lambdamucalculus forms an internal language for the
dual class of cocontrol categories. As a consequence of this
categorical duality, it follows that there is a syntactic duality
between the callbyname and the callbyvalue calculus, i.e., there
are mutually inverse translations between callbyname and
callbyvalue that preserve the categorical (and also the operational)
semantics. Such dualities have been observed in a different setting
by Filinski. In the case of the lambdamu calculus, such a syntactic
duality was conjectured by Streicher and Reus.
Comments are welcome.
Best wishes,  Peter Selinger

ABSTRACT:
We give a categorical semantics to the callbyname and callbyvalue
versions of Parigot's lambdamu calculus with disjunction types. We
introduce the class of control categories, which combine a
cartesianclosed structure with a premonoidal structure in the sense
of Power and Robinson. We prove, via a categorical structure theorem,
that the categorical semantics is equivalent to a CPS semantics in the
style of Hofmann and Streicher. We show that the callbyname
lambdamu calculus forms an internal language for control categories,
and that the callbyvalue lambdamu calculus forms an internal
language for the dual cocontrol categories. As a corollary, we obtain
a syntactic duality result: there exist syntactic translations between
callbyname and callbyvalue which are mutually inverse and which
preserve the operational semantics. This answers a question of
Streicher and Reus.
From: Philippe Gaucher
Subject: categories: Oriented homotopy and Concurrency
Date: Wed, 24 Feb 1999 23:50:21 +0100
Dear category theorists,
Here is an annoucement of preprint.
URL : http://wwwirma.ustrasbg.fr/~gaucher/multi_en.ps.gz
or http://wwwirma.ustrasbg.fr/irma/publications/1999/99010.shtml
Title : Homotopy invariants of multiple categories and concurrency in computer
science
Abstract : We associate to any (strict) multiple category $\C$ three
homology theories : the first one is called the globular homology
and it contains the oriented loops of $\C$ ; both other ones are called
corner homology, the
negative one and the positive one, which contain the corners included
in $\C$. We show up the link between this homology theories and the
homotopy of paths in multiple category. At the end of the paper, we
explain the reason why this theories are interesting for some
geometric problems coming from computer science.
Date: Thu, 25 Feb 1999 00:01:23 0500
From: "Robert A.G. Seely"
Subject: categories: Paper on linear bicategories (and noncommutative linear logic)
I would like to announce the following paper, which has been posted on
my www site . (The McGill ftp site is
currently not functionning, but as soon as it is restored, this paper
ought to appear on the Hypatia mirror site as well.)
The abstract follows.
===================================
Introduction to linear bicategories
by J.R.B. Cockett, J. Koslowski, R.A.G. Seely
Linear bicategories are a generalization of the notion of a bicategory,
in which the one horizontal composition is replaced by two (linked)
horizontal compositions. These compositions provide a semantic model for
the tensor and par of linear logic: in particular, as composition is
fundamentally noncommutative, they provides a suggestive source of
models for noncommutative linear logic.
In a linear bicategory, the logical notion of complementation becomes a
natural linear notion of adjunction. Just as ordinary adjoints are
related to (Kan) extensions, these linear adjoints are related to the
appropriate notion of linear extension.
There is also a stronger notion of complementation, which arises, for
example, in cyclic linear logic. This sort of complementation is
modelled by cyclic adjoints. This leads to the notion of a *linear
bicategory and the coherence conditions which it must satisfy. Cyclic
adjoints also give rise to linear monads: these are, essentially, the
appropriate generalization (to the linear setting) of Frobenius
algebras.
A number of examples of linear bicategories arising from different
sources are described, and a number of constructions which result in
linear bicategories are indicated.
This paper is dedicated to Jim Lambek, as part of the celebration of his
75th birthday.
Date: Tue, 9 Mar 1999 10:17:18 0500 (EST)
From: James Stasheff
Subject: categories: Quantum vertex algebras
math.QA/9903038 [abs, src, ps, other] :
Title: Quantum vertex algebras
Authors: Richard E. Borcherds
Comments: 18 pages, plain tex
Subjclass: Quantum Algebra; Category Theory
concludes with a large set of problems
some of which are strictly ncategorical
The purpose of this paper is to make the theory of vertex algebras
trivial. We do this by
setting up some categorical machinery so that vertex algebras are
just ``singular
commutative rings'' in a certain category. This makes it easy to
construct many examples of
vertex algebras, in particular by using an analogue of the
construction of a twisted group ring
from a bicharacter of a group. We also define quantum vertex
algebras as singular braided
rings in the same category and construct some examples of them. The
constructions work
just as well for higher dimensional analogues of vertex algebras.
(18kb)
************************************************************
Jim Stasheff jds@math.upenn.edu
146 Woodland Dr
Lansdale PA 19446 (215)8226707
Jim Stasheff jds@math.unc.edu
MathUNC (919)9629607
Chapel Hill NC FAX:(919)9622568
275993250
Date: Thu, 11 Mar 1999 14:20:07 +0000 (GMT)
From: Anne Heyworth
Subject: categories: Rewrite Methods for Kan Extensions of Actions of Categories
The following is available on the xxx archive.
http://xxx.soton.ac.uk/abs/math.CO/9903032
Title: Using Rewriting Systems to Compute Kan Extensions and Induced
Actions of Categories
Authors: Ronald Brown, Anne Heyworth (University Of Wales, Bangor)
Comments: 31 pages, LaTeX2e, (submitted to JSC)
Subjclass: Combinatorics
MSCclass: 68Q42 18A40 68Q40
The basic method of rewriting for words in a free monoid given a
monoid presentation is extended to rewriting for paths in a free
category given a `Kan extension presentation'. This is related to
work of CarmodyWalters on the ToddCoxeter procedure for Kan extensions,
but allows for the output data to be infinite, described by a language.
The result also allows rewrite methods to be applied in a
greater range of situations and examples, in terms of induced actions of
monoids, categories, groups or groupoids. (28kb)
Prof R. Brown,
School of Mathematics,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474office: 382475
fax: +44 1248 383663
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
New article: Higher dimensional group theory
Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm
Dr Anne Heyworth,
School of Mathematics,
University of Wales, Bangor
home page: http://www.bangor.ac.uk/~map130/welcome.html
Date: Wed, 10 Mar 1999 14:29:03 +0800 (CST)
From: farn@iis.sinica.edu.tw (Farn Wang)
Subject: categories: Report on verification of unknown number of processes
Recently, we have developed a new verification technique which
can be used to verify infinite state systems.
A manuscript:
Verification of Dynamic Linear Lists for All Numbers of Processes
in PostScript format can be obtained through my webpage:
http://www.iis.sinica.edu.tw/~farn/index.html#quo
A preliminary report can also be found in TRIIS98019 published last year.
Thank you for reading this email.
Best wishes,
Farn
ABSTRACT:
In realworld design and verification of concurrent systems
with many identical processes, the
number of processes is never a factor in the system correctness.
This paper embodies such an engineering reasoning to
propose an almost automatic method to safely verify
safety properties of such systems.
The central idea is to construct a finite collective quotient structure
(CQS) which collapses statespace representations for all system
implementations with all numbers of processes.
The problem is presented as safety bound problem which ask if
the number of processes satisfying a certain property exceeds a given bound.
Our method can be applied to systems with dynamic linear lists of unknown
number of processes.
Processes can be deleted from or inserted at any position of the linear list
during transitions.
We have used our method to develop CQS constructing algorithms for
two classes of concurrent systems :
(1) untimed systems with a global waiting queue and
(2) densetime systems with one local timer per process.
We show that our method is both sound and complete in verifying the
first class of systems.
The verification problem for the second class systems is
undecidable even with only one global binary variable.
However, our method can still automatically generate a
CQS of size no more than 1512 nodes to
verify that an algorithm in the class: Fischer's timed
algorithm indeed preserves mutual exclusion for any number of processes.
Subject: categories: Weak Bisimulation and Open Maps
Date: Thu, 06 May 1999 12:54:46 +0100
From: Luca Cattani
The following paper, to be presented at LICS '99, is available from my home
page:
http://www.cl.cam.ac.uk/~glc25/weabom.html .
Weak Bisimulation and Open Maps
Marcelo Fiore Gian Luca Cattani Glynn Winskel
COGS Computer Laboratory BRICS
Univ. Sussex, UK Univ. Cambridge, UK Univ. Aarhus, DK
Abstract
A systematic treatment of weak bisimulation and observational congruence on
presheaf models is presented. The theory is developed with respect to a
``hiding'' functor from a category of paths to observable paths. Via a view
of processes as bundles, we are able to account for weak morphisms (roughly
only required to preserve observable paths) and to derive a saturation monad
(on the category of presheaves over the category of paths). Weak morphisms
may be encoded as strong ones via the Kleisli construction associated to the
saturation monad. A general notion of weak openmap bisimulation is
introduced, and results relating various notions of strong and weak
bisimulation are provided. The abstract theory is accompanied by the concrete
study of two key models for concurrency, the interleaving model of
synchronisation trees and the independence model of labelled event structures.
Comments are welcome.
Luca


Gian Luca Cattani Phone: +44 (0)1223 334697
Cambridge University Fax: +44 (0)1223 334678
Computer Laboratory, email: Luca.Cattani@cl.cam.ac.uk
New Museums Site,
Pembroke Street,
CB2 3QG, Cambridge,
United Kingdom

Date: Fri, 7 May 1999 11:51:07 +0100
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: Preprint: Combinatorial homology and image analysis
The following preprint is available, at my home page and by ftp:
Combinatorial homology and image analysis
M. Grandis
ABSTRACT. This is the sequel of a paper, cited as Part I ("An intrinsic
homotopy theory for simplicial complexes, with applications to image
analysis"), introducing intrinsic homotopies and homotopy groups for
simplicial complexes. We study here the relations of this intrinsic
homotopy theory with the wellknown intrinsic homology theory of simplicial
complexes.
Also here, the applications are aimed at image analysis. A metric
space X has a structure t_e(X) of simplicial complex at each resolution
e > 0; the corresponding *metric combinatorial homology groups* H_n(
t_e(X)) can be directly computed, in cases of interest for applications,
via the MayerVietoris exact sequence and a study of deformation retracts
given in Part I.
http://www.dima.unige.it/STAFF/GRANDIS/
"ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/Cmb2.May99.ps
With best regards
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
email: grandis@dima.unige.it
tel: +39.010.353 6805
fax: +39.010.353 6752
Date: Wed, 16 Jun 1999 15:51:20 +0200 (MET DST)
From: "I. Moerdijk"
Subject: categories: Preprint: I. Moerdijk: "On the ConnesKreimer construction of Hopf algebras"
I. Moerdijk: "On the ConnesKreimer construction of Hopf algebras"
Abstract: We give a universal construction of families of Hopf Palgebras
for any Hopf operad P. As a special case, we recover the ConnesKreimer
Hopf algebra of rooted trees.
Available from http://www.math.uu.nl/people/moerdijk/preprints.html.

Date: Sat, 19 Jun 1999 12:35:14 0700
From: Vaughan Pratt
Subject: categories: Coimbra course notes on Chu Spaces
The notes for my course on Chu Spaces in Coimbra next month are online at
http://boole.stanford.edu/pub/coimbra.ps.gz
Vaughan Pratt
Date: Tue, 22 Jun 1999 15:53:36 +1000
From: Claudio Hermida
Subject: categories: preprint: Representable Multicategories
The preprint `Representable multicategories' is available at
http://www.maths.usyd.edu.au:8000/u/hermida
Abstract: We introduce the notion of representable multicategory,
which stands in the same relation to that of monoidal category as
fibration does to contravariant pseudofunctor (into Cat). We give an
abstract reformulation of multicategories as monads in a suitable Kleisli
bicategory of spans. We describe representability in elementary terms via
universal arrows. We also give a doctrinal characterisation of
representability based on a fundamental monadic adjunction between the
2category of multicategories and that of strict monoidal categories. The
first main result is the coherence theorem for representable
multicategories, asserting their equivalence to strict ones, which we
establish via a new technique based on the above doctrinal
characterisation. The other main result is a 2equivalence between the
2category of representable multicategories and that of monoidal
categories and strong monoidal functors. This correspondence extends
smoothly to one between bicategories and a localised version of
representable multicategories.

Claudio Hermida
School of Mathematics and Statistics F07,
University of Sydney,
Sydney, NSW 2006,
Australia
Date: Thu, 24 Jun 1999 08:53:40 0500
From: Walter Tholen
Subject: categories: HHA article available
The following paper is now available:
A Categorical Guide to Separation, Compactness and Perfectness
Walter Tholen
Based on a rather arbitrary class of morphisms in a category, which play the role of "closed
maps", we present a general approach to separation and compactness, both at the object and the
morphism levels. It covers essential parts of the classical topological theory, generalizes
various previous categorical treatments of the theme, and allows for a number of less expected
applications outside topology.
Homology, Homotopy and Applications, Vol. 1, 1999, No. 6, pp 147161
http://www.rmi.acnet.ge/hha/volumes/1999/n6/n6.dvi
http://www.rmi.acnet.ge/hha/volumes/1999/n6/n6.ps
http://www.rmi.acnet.ge/hha/volumes/1999/n6/n6.dvi.gz
http://www.rmi.acnet.ge/hha/volumes/1999/n6/n6.ps.gz
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/1999/n6/n6.dvi
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/1999/n6/n6.ps
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/1999/n6/n6.dvi.gz
ftp://ftp.rmi.acnet.ge/pub/hha/volumes/1999/n6/n6.ps.gz
Date: Mon, 12 Jul 1999 16:21:32 +0200
From: Anton Antonov
Subject: categories: Category theory for Mathematica > HPF
Hi,
Recently I presented in the Hewlett Packard conference HiPer'99 the
paper
"Translating Mathematica expressions to High Performance Fortran"
with the following abstract:
This paper introduces some ideas for translating the functional language
Mathematica to the dataparallel language High Performance Fortran
(HPF). It first discusses why we have the ability to do that. Then it
gives some interpretations by Category Theory. Third the translating
approach is presented for different Mathematica expressions that could
be interpreted as specifications for parallel independence, reduction,
task parallelism and
subprogram's data mapping. Last is shown a simple executable program
generated by the translator.
You can find more about that on
http://www.imm.dtu.dk./~aaa/MathematicaToHPF.html .
I am glad that Category theory exists! With it I was able to express
that Functional, Objectoriented and Parallel programing are the same
kind of management.
Regards
Anton
From: Peter Selinger
Subject: categories: Paper announcement
Date: Sat, 31 Jul 1999 00:45:43 0400 (EDT)
Dear Category Theorists,
I am pleased to announce the availability of a new paper,
Categorical Structure of Asynchrony,
available via http://www.math.lsa.umich.edu/~selinger/papers.html.
In this paper, I investigate properties of traced monoidal categories
that are satisfied by networks of asynchronously communicating
processes. Among these properties are Hasegawa's uniformity principle,
as well as a version of Kahn's principle: the subcategory of
*deterministic* processes is equivalent to a category of domains.
The paper also contains the following observation, which may be of
interest to categorists. I do not know whether this was observed
before, and would be grateful for references. Suppose T:Set>Set is a
functor which is lax for the symmetric monoidal structure given by
products on the category of sets. Then T associates to any category C
another category C', which Benabou called the "direct image of C by
T". This category is defined as follows:
obj(C') = obj(C), and C'(X,Y) = T(C(X,Y)).
The observation is that direct images preserve linear structure. More
precisely, if the category C possesses some algebraic structure which
is given by linear equations, then C' inherits that structure.
Nonlinear structure is not in general preserved, although one can
give conditions on T under which the construction will preserve, for
instance, affine structure. One can also loosen the conditions on T,
so that it will only preserve noncommutative linear structure.
One can use the direct image construction to extract the linear "part"
of an arbitrary algebraic structure: for instance, if C has finite
products, then C' has a monoidal structure with diagonals, which is
precisely the part of a finite product structure which is given by
linear equations.
Traced monoidal structure with diagonals is the linear part of finite
product structure with fixpoints. One direction of this, namely that
the latter structure is a special case of the former, was observed by
Hasegawa and by Hyland, but I don't know whether it had been noticed
that the former is precisely the linear part of the latter.
An example of a noncommutative linear structure (given by linear
equations where the variables occur in the same lefttoright order on
both sides) is the premonoidal structure of Power and Robinson. This
is precisely the noncommutative part of monoidal structure.
More details and examples are in the paper. Comments are, as usual,
welcome. Best wishes,  Peter Selinger

ABSTRACT:
We investigate a categorical framework for the semantics of
asynchronous communication in networks of parallel processes.
Abstracting from a category of asynchronous labeled transition
systems, we formulate the notion of a categorical model of asynchrony
as a uniformly traced monoidal category with diagonals, such that
every morphism is total and the focus is equivalent to a category of
complete partial orders. We present a simple, nondeterministic,
cpobased model that satisfies these requirements, and we discuss how
to refine this model by an observational congruence. We also present a
general construction of passing from deterministic to
nondeterministic models, and more generally, from nonlinear to
linear structure on a category.
Date: Sun, 01 Aug 1999 14:55:53 +0300
From: Zippora ArziGonczarowski
Subject: categories: Preprint announcement
The following preprint is available:

Perceive This as That  Analogies, Artificial Perception, and Category
Theory
By Zippora ArziGonczarowski
It is forthcoming in `The Annals of Mathematics and Artificial
Intelligence'.

Please email to zippie@actcom.co.il if you want to receive the
preprint.
((((((((
This paper is a continuation of the project that started with two papers
that were already announced on this list:
@article{aaa98,
Author = "Z. ArziGonczarowski and D. Lehmann",
title = "Introducing the Mathematical Category of Artificial
Perceptions",
journal = "Annals of Mathematics and Artificial Intelligence",
volume = "23",
number = "3,4",
month = "November",
pages = "267298",
publisher = "Baltzer Science Publishers",
address = "The Netherlands",
year = "1998" }
@article{bbb98,
Author = "Z. ArziGonczarowski and D. Lehmann",
title = "From Environments to RepresentationsA Mathematical Theory of
Artificial Perceptions",
journal = "Artificial Intelligence",
publisher = "Elsevier",
address = "Amsterdam",
volume = "102",
number = "2",
pages = "187247",
month = "July",
year = "1998" }
)))))))))

Dr. Zippora ArziGonczarowski
Typographics, Ltd.
46 Hehalutz St.
Jerusalem 96222
Israel
Tel:(+972)26437819 Fax:(+972)26434252 Email: zippie@actcom.co.il

Date: Wed, 15 Sep 1999 19:02:00 0400
From: Michael MAKKAI
I am announcing a paper, and enclose an (somewhat) extended abstract. The
paper is available at the site
ftp://ftp.math.mcgill.ca/pub/makkai ,
the name of the file is mltomcat.zip . It is a ZIPPED package of 8
POSTSCRIPT files. When accessed through NETSCAPE, there was no difficulty
getting it; but with ordinary ftping, we couldn't get to it. The problems
with the ftp sites here at McGill are being looked at, but they are not
solved yet.
The multitopic omegacategory of all multitopic omegacategories
by M. Makkai (McGill University)
September 2, 1999
Abstract
The paper gives two definitions: that of "multitopic omegacategory" and
that of "the (large) multitopic set of all (small) multitopic
omegacategories". It also announces the theorem that the latter is a
multitopic omegacategory. (The proof of the theorem will be contained in
a sequel to this paper.)
The work has two direct sources. One is the paper [H/M/P] (for the
references, see at the end of this abstract) in which, among others, the
concept of "multitopic set" was introduced. The other is the present
author's work on FOLDS, First Order Logic with Dependent Sorts. The
latter was reported on in [M2]. A detailed account of the work on FOLDS is
in [M3]. For the understanding of the present paper, what is contained in
[M2] suffices. In fact, section 1 of the present paper gives the
definitions of all that's needed in this paper; so, probably, there won't
be even a need to consult [M2].
The concept of multitopic set, the main contribution of [H/M/P], was, in
turn, inspired by the work of J. Baez and J. Dolan [B/D]. Multitopic sets
are a variant of opetopic sets of loc. cit. The name "multitopic set"
refers to multicategories, a concept originally due to J. Lambek [L], and
given an only moderately generalized formulation in [H/M/P]. The earlier
"opetopic set" of [B/D] is based on a concept of operad. I should say that
the exact relationship of the two concepts ("multitopic set" and "opetopic
set") is still not clarified. The main aspect in which the theory of
multitopic sets is in a more advanced state than that of opetopic sets is
that, in [H/M/P], there is an explicitly defined category Mlt of
*multitopes*, with the property that the category of multitopic sets is
equivalent to the category of Setvalued functors on Mlt, a result given a
detailed proof in [H/M/P]. The corresponding statement on opetopic sets
and opetopes is asserted in [B/D], but the category of opetopes is not
described. In this paper, the category of multitopes plays a basic role.
Multitopic sets and multitopes are described in section 2 of this paper;
for a complete treatment, the paper [H/M/P] should be consulted.
The indebtedness of the present work to the work of Baez and Dolan goes
further than that of [H/M/P]. The second ingredient of the Baez/Dolan
definition, after "opetopic set", is the concept of "universal cell". The
Baez/Dolan definition of weak ncategory achieves the remarkable feat of
specifying the composition structure by universal properties taking place
in an opetopic set. In particular, a (weak) opetopic (higherdimensional)
category is an opetopic set with additional properties ( but with no
additional data), the main one of the additional properties being the
existence of sufficiently many universal cells. This is closely analogous
to the way concepts like "elementary topos" are specified by universal
properties: in our situation, "multitopic set" plays the "role of the
base" played by "category" in the definition of "elementary topos". In
[H/M/P], no universal cells are defined, although it was mentioned that
their definition could be supplied without much difficulty by imitating
[B/D]. In this paper, the "universal (composition) structure" is supplied
by using the concept of FOLDSequivalence of [M2].
In [M2], the concepts of "FOLDSsignature" and "FOLDSequivalence" are
introduced. A (FOLDS) signature is a category with certain special
properties. For a signature L , an *Lstructure* is a Setvalued functor
on L. To each signature L, a particular relation between two variable
Lstructures, called Lequivalence, is defined. Two Lstructures M, N, are
Lequivalent iff there is a socalled Lequivalence span M<P>N
between them; here, the arrows are ordinary natural trasnformations,
required to satisfy a certain property called "fiberwise surjectivity".
The slogan of the work [M2], [M3] on FOLDS is that *all meaningful
properties of Lstructures are invariant under Lequivalence*. As with all
slogans, it is both a normative statement ("you should not look at
properties of Lstructures that are not invariant under Lequivalence"),
and a statement of fact, namely that the "interesting" properties of
Lstructures are in fact invariant under Lequivalence. (For some slogans,
the "statement of fact" may be false.) The usual concepts of "equivalence"
in category theory, including the higher dimensional ones such as
"biequivalence", are special cases of Lequivalence, upon suitable, and
natural, choices of the signature L; [M3] works out several examples of
this. Thus, in these cases, the slogan above becomes a tenet widely held
true by category theorists. I claim its validity in the generality stated
above.
The main effort in [M3] goes into specifying a language, First Order Logic
with Dependent Sorts, and showing that the first order properties
invariant under Lequivalence are precisely the ones that can be defined
in FOLDS. In this paper, the language of FOLDS plays no role. The concepts
of "FOLDSsignature" and "FOLDSequivalence" are fully described in
section 1 of this paper.
The definition of *multitopic omegacategory* goes, in outline, as
follows. For an arbitrary multitope SIGMA of dimension >=2, for a
multitopic set S, for a pasting diagram ALPHA in S of shape the domain of
SIGMA and a cell a in S of the shape the codomain of SIGMA, such that a
and ALPHA are parallel, we define what it means to say that a is a
*composite* of ALPHA. First, we define an auxiliary FOLDS signature
L extending Mlt, the signature of multitopic sets. Next, we define
structures S and S, both of the signature L, the first
constructed from the data S and a , the second from S and ALPHA, both
structures extending S itself. We say that a is a composite of ALPHA if
there is a FOLDSequivalencespan E between S and S that
restricts to the identity equivalencespan from S to S . Below, I'll refer
to E as an *equipment* for a being a composite of ALPHA. A multitopic
set is a *mulitopic omegacategory* iff every pasting diagram ALPHA in it
has at least one composite.
The analog of the universal arrows in the Baez/Dolan style definition is
as follows. A *universal arrow* is defined to be an arrow of the form
b:ALPHA> a where a is a composite of ALPHA via an equipment E that
relates b with the identity arrow on a : in turn, the identity arrow on
a is any composite of the empty pasting diagram of dimension dim(a)+1
based on a . Note that the main definition does *not* go through first
defining "universal arrow".
A new feature in the present treatment is that it aims directly at weak
*omega*categories; the finite dimensional ones are obtained as truncated
versions of the full concept. The treatment in [B/D] concerns finite
dimensional weak categories. It is important to emphasize that a
multitopic omega category is still just a multitopic set with additional
properties, but with no extra data.
The definition of "multitopic omegacategory" is given is section 5; it
uses sections 1, 2 and 4, but not section 3.
The second main thing done in this paper is the definition of MltOmegaCat.
This is a particular large multitopic set. Its definition is completed
only by the end of the paper. The 0cells of MltOmegaCat are the samll
multitopic omegacategories, defined in section 5. Its 1cells, which we
call 1transfors (thereby borrowing, and altering the meaning of, a term
used by Sjoerd Crans [Cr]) are what stand for "morphisms", or "functors",
of multitopic omegacategories. For instance, in the 2dimensional case,
multitopic 2categories correspond to ordinary bicategories by a certain
process of "cleavage", and the 1transfors correspond to homomorphisms of
bicategories [Be]. There are ndimensional transfors for each n in N . For
each multitope (that is, "shape" of a higher dimensional cell) PI, we
have the *PItransfors*, the cells of shape PI in MltOmegaCat.
For each fixed multitope PI, a PItransfor is a *PIcolored multitopic
set* with additional properties. "PIcolored multitopic sets" are defined
in section 3; when PI is the unique zerodimensional multitope, PIcolored
multitopic sets are the same as ordinary multitopic sets. Thus, the
definition of a transfor of an arbitrary dimension and shape is a
generalization of that of "multitopic omegacategory"; the additional
properties are also similar, they being defined by FOLDSbased universal
properties. There is one new element though. For dim(PI)>=2 , the concept
of PItransfor involves a universal property which is an
omegadimensional, FOLDSstyle generalization of the concept of right
Kanextension (right lifting in the terminology used by Ross Street).
This is a "rightadjoint" type universal property, in contrast to the
"leftadjoint" type involved in the concept of composite (which is a
generalization of the usual tensor product in modules).
The main theorem, stated but not proved here, is that MltOmegaCat is a
multitopic omegacategory.
The material in this paper has been applied to give formulations of
omegadimensional versions of various concepts of homotopy theory;
details will appear elesewhere.
I thank Victor Harnik and Marek Zawadowski for many stimulating
discussions and helpful suggestions. I thank the members of the Montreal
Category Seminar for their interest in the subject of this paper, which
made the exposition of the material at a time when it was still in an
unfinished state a very enjoyable and useful process for me.
References:
[B/D] J. C. Baez and J. Dolan, Higherdimensional algebra III.
ncategories and the algebra of opetopes. Advances in Mathematics 135
(1998), 145206.
[Be] J. Benabou, Introduction to bicategories. In: Lecture Notes in
Mathematics 47 (1967), 177 (SpringerVerlag).
[Cr] S. Crans, Localizations of transfors. Macquarie Mathematics
Reports no. 98/237.
[H/M/P] C. Hermida, M. Makkai and J. Power, On weak higher dimensional
categories I. Accepted by: Journal of Pure and Applied Algebra. Available
electronically (when the machines work ...).
[L] J. Lambek, Deductive systems and categories II. Lecture Notes in
Mathematics 86 (1969), 76122 (SpringerVerlag).
[M2] M. Makkai, Towards a categorical foundation of mathematics. In:
Logic Colloquium '95 (J. A. Makowski and E. V. Ravve, editors). Lecture
Notes in Logic 11 (1998) (SpringerVerlag).
[M3] M. Makkai, First Order Logic with Dependent Sorts. Research
momograph, accepted by Lecture Notes in Logic (SpringerVerlag). Under
revision. Original form available electronically (when the machines
work ...).
Cheers: M. Makkai
Date: Thu, 16 Sep 1999 12:15:16 0400
From: Michael MAKKAI
Subject: categories: announcement update
This is an update on the announcement I made yesterday. I announced the
paper "The multitopic omegacategory of all multitopic omegacategories".
The site I named for it does not seem to work, however. Instead, try
http://mystic.biomed.mcgill.ca/M_Makkai
As I said earlier, you'll find MLTOMCAT.ZIP, a ZIPped package of 8
POSTSCRIPT files.
Good luck: M. Makkai
Date: Sun, 19 Sep 1999 18:26:36 0400
From: Michael MAKKAI
Subject: categories: second update
This is the second update on the paper "The multitopic omegacategory of
all multitopic omegacategories". The zipfile that got onto both sites I
mentioned before was bad. I have now replaced it with another one which I
had tested for "unzipping". The two sites:
ftp://ftp.math.mcgill.ca/pub/makkai
http://mystic.biomed.mcgill.ca/M_Makkai
The filename is MLTOMCAT.ZIP. It is a ZIPped package of 8 POSTSCRIPT
files.
M. Makkai
Date: Fri, 24 Sep 1999 10:27:48 +0100
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: preprint: Simplicial toposes and combinatorial homotopy
The following preprint is available:
M. Grandis
Simplicial toposes and combinatorial homotopy,
Dip. Mat. Univ. Genova, Preprint 400 (1999).
Abstract. The term *combinatorial topos* denotes here a topos of presheaves
over a small subcategory of the category of finite sets. The main instances
we want to consider are the presheaf categories of simplicial sets, cubical
sets, and globular sets, together with their symmetric versions: e.g., the
topos !Smp of symmetric simplicial sets consists of all presheaves on the
category !Delta of finite, positive cardinals.
We show here how combinatorial homotopy, developed in previous works
for simplicial complexes (the cartesian closed subcategory of *simple*
presheaves in !Smp) can be extended to the topos !Smp. As a crucial
advantage, the (extended) fundamental groupoid Pi_1: !Smp > Gpd is left
adjoint to a natural functor M_1: Gpd > !Smp, the symmetric nerve of a
groupoid, and therefore  as a strong van Kampen property  preserves all
colimits.
Analogously, a notion of (nonreversible) *directed* homotopy can be
developed in Smp, with applications to image analysis similar to the ones
of the symmetric case. We have now a homotopy ncategory functor C_n: Smp
> nCat, left adjoint to a nerve N_n = nCat(C_n(Delta[n]), ). It
would be interesting to determine whether the ncategory C_n(Delta[n])
coincides with Street's oriental O_n, and the previous nerve with
Street's, as it seems likely.
___
Available at:
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/CmbTop.Sep99.ps
(459 K)
___
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
email: grandis@dima.unige.it
tel: +39.010.353 6805 fax: +39.010.353 6752
http://www.dima.unige.it/STAFF/GRANDIS/
ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/
Date: Fri, 01 Oct 1999 18:30:06 +0000
From: Fabio Gadducci
Subject: categories: paper announcement
Dear members of the mailing list, I'm pleased to annouce that the paper
``Rewriting on Cyclic Structures'',
by myself and Andrea Corradini, is available at
http://www.di.unipi.it/~gadducci/papers/RAIRO.ps.
The abstract follows, but shortly, it uses traced monoidal 2categories
where, in addition, each object ha`s a comonoidal structure in order
to simulate various kinds of (eventually cyclic) term (graph) rewriting.
Its interest for a broader audience may lie, besides in showing a
practical application of the trace structure in the rewriting field, in
its appendix, where we tried to sketch a very SHORT history of the
notion of feedback in theoretical computer science, with a particular
attention to the algebraic specification field. We found it interesting
to review previous approaches to the topic, after the results of
JoyalStreetVerity have newly sparkled the interest in the algebraic
description of fixed points (see e.g. the recent paper by Selinger
advertised a few weeks ago on this mailing list).
Best regards,
Fabio Gadducci
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
\begin{abstract}
We present a categorical formulation of the rewriting of possibly cyclic
term graphs, based on a variation of algebraic 2theories. We show that
this presentation is equivalent to the wellaccepted operational
definition proposed by Barendregt et aliibut for the case of
``circular redexes'', for which we propose (and justify formally) a
different treatment. The categorical framework allows us to model in a
concise way
also automatic garbage collection and rules for sharing/unsharing and
folding/unfolding of structures, and to relate term graph rewriting to
other rewriting formalisms.
\end{abstract}
Date: Tue, 12 Oct 1999 14:18:48 +0100
From: kock
Subject: categories: preprint available
The preprint
Algebra of Principal Fibre Bundles, and Connections
is available at
ftp://ftp.imf.au.dk/pub/kock/princ4.ps
(102 kb).
The classical relationship between the curvature of a connection,
and the coboundary of its connection form, here comes about from a
pure groupoid calculation.
The preprint updates and expands my 1983/1986 paper,
"Combinatorial notions relating to principal fibre bundles".
Anders Kock
http://www.imf.au.dk/~kock/
From: Martin Escardo
Date: Fri, 29 Oct 1999 16:25:57 +0100 (BST)
Subject: categories: function spaces
Dear Comprox and Categories members,
Here is a short note that Reinhold Heckmann and I have written. Your
comments are welcome, as always.
On function spaces in topology

It is the purpose of this expository note to provide a
selfcontained, elementary and brief development of the fact that
the exponentiable topological spaces are precisely the
corecompact spaces. The only prerequisite is a basic knowledge of
topology (continuous functions, product topology and compactness).
We hope that teachers and students of topology will find this
useful. As far as we know, there is no such development available
in the literature. Although there are one or two embellishments,
our methods are certainly not original. We briefly discuss more
advanced treatments in the introduction.

http://www.dcs.stand.ac.uk/~mhe/papers/exponentiablespaces.ps.gz
http://www.dcs.stand.ac.uk/~mhe/papers/exponentiablespaces.dvi.gz
http://www.dcs.stand.ac.uk/~mhe/papers/exponentiablespaces.ps
http://www.dcs.stand.ac.uk/~mhe/papers/exponentiablespaces.dvi

Best regards,
Martin & Reinhold
Subject: categories: Paper announcement
Date: Fri, 12 Nov 1999 14:05:07 +0000
From: Luca Cattani
The following paper is available at
http://www.cl.cam.ac.uk/~glc25/premcl.html .
It will also be available soon as BRICS Report, RS9936 (see
www.brics.dk/Publications), and as Cambridge University Computer Laboratory
Technical Report n. 477 (contact techreports@cl.cam.ac.uk to obtain a hard
copy) :
Presheaf Models for CCSlike Languages
Gian Luca Cattani Glynn Winskel
Computer Laboratory BRICS
University of Cambridge University of Aarhus
England Denmark
Abstract
=========
The aim of this paper is to harness the mathematical machinery around
presheaves for the purposes of process calculi. Joyal, Nielsen and Winskel
proposed a general definition of bisimulation from open maps. Here we show
that openmap bisimulations within a range of presheaf models are congruences
for a general process language, in which CCS and related languages are easily
encoded. The results are then transferred to traditional models for processes.
By first establishing the congruence results for presheaf models, abstract,
general proofs of congruence properties can be provided and the awkwardness
caused through traditional models not always possessing the cartesian
liftings, used in the breakdown of process operations, are sidestepped. The
abstract results are applied to show that hereditary historypreserving
bisimulation is a congruence for CCSlike languages to which is added a
refinement operator on event structures as proposed by van Glabbeek and Goltz.
Date: Mon, 15 Nov 1999 08:49:41 +1100
From: Claudio Hermida
Subject: categories: coherence => universality (preprint)
The preprint "From coherent structures to universal properties"
is available from
http://www.maths.usyd.edu.au:8000/u/hermida
under cohuniv.ps
Abstract: Given a 2category K admitting a calculus of bimodules, and a
2monad T on it compatible with such calculus, we construct a 2category
L with a 2monad S on it such that: i) S has the adjointpseudoalgebra
property.
ii) The 2categories of pseudoalgebras of S and T are equivalent.
Thus, coherent structures (pseudoTalgebras) are transformed into
universally characterised ones (adjointpseudoSalgebras). The
2category L consists of lax algebras for the pseudomonad induced by T
on the bicategory of bimodules of K. We give an intrinsic
characterisation of pseudoSalgebras in terms of {\em
representability\/}. Two major consequences of the above transformation
are the classifications of lax and strong morphisms, with the attendant
coherence result for pseudoalgebras. We apply the theory in the context
of internal categories and examine monoidal and monoidal globular
categories (including their {\em monoid classifiers\/}) as well as
pseudofunctors into Cat.

Claudio Hermida
School of Mathematics and Statistics F07,
University of Sydney,
Sydney, NSW 2006,
Australia
From: Martin Escardo
Date: Thu, 18 Nov 1999 09:15:52 +0000 (GMT)
Subject: categories: Re: function spaces
I wrote:
> It is the purpose of this expository note to provide a
> selfcontained, elementary and brief development of the fact that
> the exponentiable topological spaces are precisely the
> corecompact spaces. The only prerequisite is a basic knowledge of
> topology (continuous functions, product topology and compactness).
> We hope that teachers and students of topology will find this
> useful. As far as we know, there is no such development available
> in the literature. Although there are one or two embellishments,
> our methods are certainly not original.
>
>
> http://www.dcs.stand.ac.uk/~mhe/papers/exponentiablespaces.ps
It turns out, as Fred Linton kindly let me know just after I posted
this, that Eilenberg developed such an account to general function
spaces in topology. Yesterday I got a copy of Eilenberg's manuscript
(in the literal sense of manuscript) that Fred Linton sent me, which I
read with pleasure. Apparently this will be eventually published. It
was written around 1985. So, after all, there is (going to be) such a
development available in the literature.
The methods that both papers use are the same, and are due to Fox,
Arens, Dugundji, Day and Kelly, Scott, and Isbell (although we combine
them in different ways). These references and most of these methods
are discussed in a paper on function spaces published by Isbell in
1985.
Martin