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Date: Tue, 2 Nov 2004 18:14:52 0400 (AST)
From: Bob Rosebrugh
To: categories
Subject: categories: Many thanks
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There are about 25 volunteers to work on the Max Kelly book. I'll be in
touch with some of you about typing shortly, and the others a little
later. For now, many thanks to all those who have volunteered. We should
be able to have the book freely available in a matter of weeks.
Bob Rosebrugh
3Nov2004 09:45:08 0400,2440;00000000000100000000
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Date: Mon, 01 Nov 2004 11:21:31 0500
From: Michael A Warren
To: categories@mta.ca
Subject: categories: Thesis available
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To Whom It May Concern,
I would like to announce the availability of my recently completed Master's
thesis (written under the supervision of Steve Awodey) on the algebraic set
theory website:
http://www.phil.cmu.edu/projects/ast/
The title and abstract are included below in plain text.
Best regards,
Michael Warren

Title: Predicative Categories of Classes.
Abstract:
In this thesis the tools of category theory and categorical logic are
employed in order to study predicative set theories. Specifically, we
introduce two constructive set theories BCST and CST and prove that they
are sound and complete with respect to models in categories with certain
structure. Specifically, _basic categories of classes_ and _categories of
classes_ are axiomatized and shown to provide models of the aforementioned
set theories.
We then show that given any Heyting pretopos E there exists a subcategory
Idl(E) of sheaves on E, called the _ideal completion of_ E, which is such a
category of classes. Specifically, we construct fixed points for the
powerobject functor P(): Idl(E) > Idl(E) in order to build models of the
untyped set theory BCST. Furthermore, if E is a locally cartesian closed
pretopos, then the construction yields models of CST inside Idl(E).
Finally, it is a consequence of this work that the set theories in question
are sound and complete with respect to such models in categories of ideals.
This embedding results serves to establish, in effect, the conservativity
of the set theory CST over a form of dependent type theory. Additional
minor results which may (or may not) be of independent interest are also
obtained regarding the categories in question.
5Nov2004 20:56:13 0400,2369;00000000000100000000
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Date: Thu, 4 Nov 2004 15:34:10 +0000 (GMT)
From: Paul B Levy
To: categories@mta.ca
Subject: categories: adjunctions for callbypushvalue
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Dear all,
My paper "Adjunction models for CallByPushValue with stacks", to appear
in Theory and Applications of Categories (the special edition for CTCS
2002), is on my webpage
http://www.cs.bham.ac.uk/~pbl/papers/
The abstract is below.
regards
Paul
Callbypushvalue is a "semantic machine code", providing a set of simple
primitives from which both the callbyvalue and callbyname paradigms
are built. We present its operational semantics as a stack machine,
suggesting a term judgement of stacks. We then see that CBPV,
incorporating these stack terms, has a simple categorical semantics based
on an adjunction between values and stacks. There are no coherence
requirements.
We describe this semantics incrementally. First, we introduce locally
indexed categories and the opGrothendieck construction, and use these to
give the basic structure for interpreting the three judgements: values,
stacks and computations. Then we look at the universal property required
to interpret each type constructor. We define a model to be a strong
adjunction with countable coproducts, countable products and exponentials.
We see a wide range of instances of this structure: we give examples for
divergence, storage, erratic choice, continuations, possible worlds and
games (with or without a bracketing condition), in each case resolving the
strong monad from the literature into a strong adjunction. And we give
ways of constructing models from other models.
Finally, we see that callbyvalue and callbyname are interpreted within
the Kleisli and coKleisli parts, respectively, of a callbypushvalue
adjunction.
6Nov2004 11:30:42 0400,2363;00000000000000000000
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XOrganisation: Faculty of Science, University of Amsterdam, The Netherlands
XURL: http://www.science.uva.nl/
Subject: categories: ESSLLI'05 Student Session Call for Papers
From: "Judit Gervain"
Date: Wed, November 3, 2004 14:28
To: info@folli.org
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We apologize for multiple postings of this call.

1st Call for Papers: ESSLLI'05 Student Session
We are pleased to announce the Student Session (StuS) of the 17th
European Summer School in Logic, Language and Information (ESSLLI'05,
819 August, Edinburgh, http://www.macs.hw.ac.uk/esslli05/). We invite
papers for oral and poster presentation from the areas of Logic,
Language and Computation.
The ESSLLI Student Session encourages submissions from students at any
level, undergraduate, as well as postgraduate. This year, unlike in the
past, papers can be submitted for oral OR poster presentation
separately.
Student authors are invited to submit a full paper, not to exceed 7 pages
of length exclusive of references. Papers are to be submitted with clear
indications of the selected modality of presentation, i.e. oral or
poster. The submissions will be reviewed by the student session program
committee and selected reviewers.
The preferred formats of submissions are PostScript, PDF, or plain text,
although other formats will also be accepted.
The paper and a separate identification page must be sent
electronically to: gervain@sissa.it.
Deadline: 15th February 2005.
For more information and the technical details of the submission, see:
http://www.sissa.it/~gervain/StuS.html or write to: gervain@sissa.it
========================@========================
Judit Gervain
SISSA CNS
via Beirut 24
34014 Trieste
Italy
office tel: +39 040 37 87 613
mobile tel: +39 329 788 40 25
www.sissa.it/~gervain
10Nov2004 12:16:52 0400,7526;00000000000000000000
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Subject: categories: Call for Papers: FM 2005
To: events@fmeurope.org
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Call for Papers: FM 2005
The 13th International Symposium of Formal Methods Europe
FM'05  Newcastle upon Tyne, UK
1822 July 2005
www.csr.ncl.ac.uk/fm05/
Important dates:
24 January 2005 Paper submission deadline
07 March 2005 Last date for Workshop & Tutorial proposals
09 April 2005 Decisions on papers
02 May 2005 Final versions of papers due
09 May 2005 Last date for Tools Exhibition & Demonstration proposal=
s
FM'05 is the thirteenth in a series of symposia organized by Formal
Methods Europe,
an independent association whose aim is to stimulate the use of, and
research on,
formal methods for software development. The symposia have been notably
successful
in bringing together innovators and practitioners in precise mathematical
methods
for software development, industrial users as well as researchers.
Submissions will
be welcomed in the form of original papers on research and practice,
proposals for
workshops and tutorials, and entries for the exhibition of software tools=
,
publications and companies.
FM'05 welcomes papers in all aspects of formal methods for computer syste=
ms,
including, but not restricted to, the following:
 introducing formal methods in industrial practice (technical,
organizational,
social, psychological aspects)
 reports on practical use and case studies (reporting positive or negati=
ve
experiences)
 formal methods in hardware and system design
 reusable domain theories
 theoretical foundations (specification and modelling, refining,
verification,
calculation etc.)
 tool support and software engineering
 environments for formal methods
 method integration
Papers
=3D=3D=3D=3D=3D=3D
Full papers should be submitted electronically via the Web by 24 January
2005. Full
submission details will be published on the conference web site. Papers
will be
evaluated by the Program Committee according to their originality,
significance,
soundness, quality of presentation and relevance with respect to the main
issues of
the symposium. Papers should have not been submitted elsewhere for
publication.
Accepted papers will be published in the Symposium Proceedings, to appear=
in
SpringerVerlag's Lecture Notes in Computer Science Series. Papers should
not exceed
16 pages and should be in LNCS format: see
http://www.springer.de/comp/lncs/authors.html
Please include a short list of keywords on a separate line at the end of =
the
abstract, beginning with "Keywords:" in boldface.
Workshops
=3D=3D=3D=3D=3D=3D=3D=3D=3D
We welcome proposals for oneday or
twoday workshops related to FM'05. Proposals may be considered and
evaluated at
any time up to 7 March 2005 and should be directed to the Workshop Chair
Juan
Bicarregui (J.C.Bicarregui@rl.ac.uk). Early contact is recommended.
Tutorials
=3D=3D=3D=3D=3D=3D=3D=3D=3D
We welcome proposals for halfday or full
day tutorials related to formal methods.
Tutorial proposals will be evaluated on the basis of their potential
benefit for
participants, and should contain an outline of the objectives, format,
content and,
if appropriate, history of the tutorial. Proposals may be considered and
evaluated
at any time up to 7 March 2005, and should be sent to the Tutorial Chair,=
=20
Neil
Henderson (Neil.Henderson@ncl.ac.uk). Early contact is recommended.
Exhibition & Sponsors' Presentations
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
An exhibition of projects and tools will also take place during the
symposium, with
the opportunity of holding presentations for each tool. The opportunity
also exists
to give a presentation as a sponsor. Proposals are welcomed at any time
up to 9 May
2005 and should be directed to the Exhibition and Sponsors' Chair, Joan
Atkinson
(Joan.Atkinson@ncl.ac.uk). Early contact is recommended.
Contacts
General Chair: John Fitzgerald, University of Newcastle upon Tyne, UK=20
John.Fitzgerald@ncl.ac.uk
Programme Chairs:
Ian Hayes, University of Queensland, Australia
Ian.Hayes@itee.uq.edu.au
Andrzej Tarlecki, Warsaw University, Poland
tarlecki@mimuw.edu.pl
Organisers:
Claire Smith, University of Newcastle upon Tyne, UK
Claire.Smith@ncl.ac.uk
Jon Warwick, University of Newcastle upon Tyne, UK
Jon.Warwick@ncl.ac.uk
Workshops Chair: Juan Bicarregui, Rutherford Appleton Laboratory, UK
J.C.Bicarregui@rl.ac.uk
Tutorials Chair: Neil Henderson, University of Newcastle upon Tyne, UK
Neil.Henderson@ncl.ac.uk
Exhibitions & Sponsorship: Joan Atkinson, University of Newcastle, UK
Joan.Atkinson@ncl.ac.uk
Programme Committee
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
Bernhard Aichernig, UNUIIST, UN
Keijiro Araki, Kyushu University, Japan
Michel Bidoit, LSV, CNRS & ENS de Cachan, France
Juan Bicarregui, Rutherford Appleton Laboratory, UK
Ed Brinksma, University of Twente, The Netherlands
Luca Cardelli, Microsoft Research, UK
Ernie Cohen, Microsoft, USA
Jin Song Dong, National University of Singapore, Singapore
Jose Fiadeiro, University of Leicester, UK
John S. Fitzgerald, Centre for Software Reliability, UK
Stefania Gnesi, CNR, Italy
Anthony Hall, UK
Ian Hayes, University of Queensland, Australia (Cochair)
Anne E. Haxthausen, Technical University of Denmark, Denmark
Thomas Henzinger, University of California, Berkeley, USA
He Jifeng, UNUIIST, UN
Cliff Jones, University of Newcastle upon Tyne, UK
Shaoying Liu, Hosei University, Japan
M=EDche=E1l Mac an Airchinnigh, Trinity College, Dublin, Ireland
Tom Maibaum, McMaster University, Canada
Dino Mandrioli, Politecnico di Milano, Italy
Tobias Nipkow, Technische Universit=E4t M=FCnchen, Germany
Jos=E9 Oliveira, Universidade do Minho, Portugal
Sam Owre, CRI, USA
Alexander Petrenko, ISPRAS, Russia
Nico Plat, West Consulting, Netherlands
Ken Robinson, University of New South Wales, Australia
Mark Saaltink, ORA, Canada
Shin Sahara, JFITS, Japan
Steve Schneider, University of Surrey, UK
Kaisa Sere, =C5bo Akademi, Finland
Ketil St=F8len, SINTEF, Norway
Andrzej Tarlecki, Warsaw University, Poland (Cochair)
Martyn Thomas, Martyn Thomas Associates, UK
Mark Utting, Waikato University, New Zealand
Marcel Verhoef, Chess IT & Radboud University, Nijmegen, Netherlands Alan
Wassyng, McMaster University, Canada
Martin Wirsing, LudwigMaximiliansUniversit=E4t, M=FCnchen, Germany
 Nico Plat  mail@nicoplat.com
_______________________________________________
events mailing list
events@fmeurope.org
http://www.fmeurope.org/mailman/listinfo/events
10Nov2004 12:16:52 0400,1208;00000000000000000000
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MessageID: <418F9905.AD4DE552@maths.uct.ac.za>
Date: Mon, 08 Nov 2004 18:04:21 +0200
From: George Janelidze
Organization: University of Cape Town
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Subject: categories: CT2004 Proceedings
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Dear Participants of CT2004,
We would like to inform you that the deadline for the submission of
papers to CT2004 Proceedings Volume of "Theory and Applications of
Categories" has been extended to December 31, 2004.
Please submit your papers to any of us, in the form within the
standards required by the journal
George Janelidze, janelidg@maths.uct.ac.za
John MacDonald, johnm@math.ubc.ca
Ross Street, street@ics.mq.edu.au
Walter Tholen, tholen@mathstat.yorku.ca
11Nov2004 14:16:32 0400,1361;00000000000000000000
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Date: Thu, 11 Nov 2004 17:21:30 +0000
To: categories@mta.ca
From: Jorge Picado
Subject: categories: PSSL in Coimbra
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============================================
FIRST ANNOUNCEMENT
============================================
81th PERIPATETIC SEMINAR ON SHEAVES AND LOGIC
University of Coimbra, Coimbra, Portugal
It is proposed to hold the 81th meeting of the Peripatetic Seminar on
Sheaves and Logic in Coimbra during the weekend of 9/10 April 2005.
Further details of the arrangements will be announced by midJanuary;
in the meantime, any queries may be addressed to the organizers
Manuela Sobral ,
Maria Manuel Clementino ,
Jorge Picado or
Lurdes Sousa .
November 11, 2004
17Nov2004 09:34:19 0400,5173;00000000000100000000
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XRAVAntiVirus: This email has been scanned for viruses on host: johann.math.tulane.edu
MessageID: <41991017.3000009@math.tulane.edu>
Date: Mon, 15 Nov 2004 15:22:47 0500
From: Michael Mislove
ReplyTo: mwm@math.tulane.edu
Organization: Tulane University
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To: categories@mta.ca
Subject: categories: MFPS XXI First Call for Papers
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Dear Colleagues,
Below is the First Announcement and Call for Papers for MFPS 21, which
will take place at the University of Birmingham, UK from Wednesday, May
18 through Saturday, May 21, 2005.
Best regards,
Mike Mislove
===============================================
Professor Michael Mislove Phone: +1 504 8623441
Department of Mathematics FAX: +1 504 8655063
Tulane University URL: http://www.math.tulane.edu/~mwm
New Orleans, LA 70118 USA
===============================================
First Announcement and Call for Papers
MFPS XXI
Twentyfirst Conference on the
Mathematical Foundations of Programming Semantics
University of Birmingham
Edgbaston, Birmingham UK
May 18  May 21, 2005
Partially Supported by US Office of Naval Research
The Twentyfirst Conference on the Mathematical Foundations of
Programming Semantics will take place at the University of
Birmingham, UK from Wednesday, May 18 through Saturday, May 21, 2005.
The invited speakers for MFPS XXI are
Samson Abramsky, Oxford
Andrej Bauer, IMFM, Slovenia
Cliff Jones, Newcastle
Catuscia Palamidessi, INRIA
Gordon Plotkin*, Edinburgh
John Reynolds, CMU
*: To be confirmed
There also will be a plenary talk on security. In addition to the
invited addresses, there will be three special sessions:
o Special Session on Quantum Computing organized by Samson Abramsky,
Michael Mislove (Tulane) and Prakash Panangaden (McGill).
o Special Session on Security organized by Catherine Meadows (NRL)
o Special Session on Domain Theory and Topology, organized by
Martin Escardo and Achim Jung (Birmingham).
The remainder of the program will be composed of papers selected by the
Program Committee from submissions received in response to this Call for
Papers. The Program Committee is being chaired by Martin Escardo
(Birmingham). It also includes:
o Ulrich Berger, Swansea
o Lars Birkedal, ITU, Denmark
o Stephen Brookes, CMU
o Thierry Coquand, Goteberg
o PierreLouis Curien, LIAFA, Paris VII
o Vincent Danos, Paris
o Marcelo Fiore, Cambridge
o Achim Jung Birmingham, U.K.
o Catherine Meadows, NRL
o Michael Mislove, Tulane
o Luke Ong, Oxford
o Prakash Panangaden, McGill
o Brigitte Pientka, McGill
o Phil Scott, Ottawa
o Roberto Segala, Verona
0 Alex Simpson, Edinburgh
Submissions should consist of original work that has not been published
elsewhere. Submissions should be no longer than 12 pages, and they
should be in the form of either PostScript or pdf files that can be
printed on a standard printer. They can be made using the link that will
be available on the MFPS 21 Home Page
http://www.math.tulane.edu/~mfps/mfps21.htm  submissions will open in
early January.
Submissions must be received by midnight, Pacific Standard Time on
Friday, February 15, 2005.
Authors will be notified of acceptance by March 25, 2005.
The MFPS conferences are devoted to those areas of mathematics, logic
and computer science which are related to the semantics of programming
languages. The series particularly has stressed providing a forum where
both mathematicians and computer scientists can meet and exchange ideas
about problems of common interest. We also encourage participation by
researchers in neighboring areas, since we strive to maintain breadth in
the scope of the series.
The Organizing Committee for MFPS consists of Stephen Brookes (CMU),
Achim Jung (Birmingham), Catherine Meadows (NRL), Michael Mislove
(Tulane) and Prakash Panangaden (McGill). The local arrangements for
MFPS XXI are being overseen by Achim Jung.
In addition to supporting the conference overall, the support we
anticipate from the Office of Naval Research makes funds available to
help offset expenses of graduate students. Women and minorities also are
encouraged to inquire about possible support to attend the meeting.
Participation Information
Information about MFPS XXI can be found at the URL
http://www.math.tulane.edu/~mfps/mfps21.htm Registration information
will be available at these sites shortly after the New Year. If you have
problems accessing the link above, then send email to mfps@math.tulane.edu.
19Nov2004 08:32:04 0400,2631;00000000000100000000
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Subject: categories: Three papers on selfsimilarity
From: Tom Leinster
To: categories@mta.ca
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This is to announce the availability of three papers giving a general
theory of selfsimilarity. The first is an informal overview; the
second two are the serious business. The seed was Peter Freyd's
universal characterization of the real interval.
In doing this research I've had a lot of help from people in response to
questions that I've posted on this list, and of course this is also
where Peter's original result appeared. I'd therefore like to express
publicly my thanks to Bob Rosebrugh for his work in running it.
Best wishes, Tom
1. "General selfsimilarity: an overview"
Informal seminar notes explaining the ideas in (2) and (3).
http://arxiv.org/abs/math.DS/0411343
2. "A general theory of selfsimilarity I"
Consider a selfsimilar space X. A typical situation is that X looks
like several copies of itself glued to several copies of another space
Y, and Y looks like several copies of itself glued to several copies of
X, or the same kind of thing with more than two spaces. Thus, the
selfsimilarity of X is described by a system of simultaneous
equations. Here I formalize this idea and the notion of a `universal
solution' of such a system. I determine exactly when a system has a
universal solution and, when one does exist, construct it.
http://arxiv.org/abs/math.DS/0411344
3. "A general theory of selfsimilarity II: recognition"
This paper concerns the selfsimilarity of topological spaces, in the
sense defined in (2). I show how to recognize selfsimilar spaces, or
more precisely, universal solutions of selfsimilarity systems.
Examples include the standard simplices (selfsimilar by barycentric
subdivision) and solutions of iterated function systems. Perhaps
surprisingly, every compact metrizable space is selfsimilar in at least
one way. From this follow the classical results on the role of the
Cantor set among compact metrizable spaces.
http://arxiv.org/abs/math.DS/0411345
19Nov2004 08:32:04 0400,1231;00000000000000000000
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Date: Thu, 18 Nov 2004 08:44:42 +1100 (EST)
MessageId: <200411172144.iAHLigSO028393@cooper.uws.edu.au>
From: "Stephen Lack"
To: categories@mta.ca
Subject: categories: Conference for 60th birthday of Ross Street
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This is a preliminary announcement concerning a conference
Categories in Algebra, Geometry, and Mathematical Physics
to mark the 60th birthday of Ross Street. The conference will
be held at Macquarie University in Sydney, Australia, during
the period 1115 July 2005.
The conference website is at
http://streetfest.maths.mq.edu.au/
and includes a preliminary list of participants. More information
will be provided in due course.
Michael Batanin
Alexei Davydov
Mike Johnson
Steve Lack
Amnon Neeman
(the organizing committee)
22Nov2004 16:52:19 0400,1395;00000000000000000000
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Date: Mon, 22 Nov 2004 14:27:58 +0000
From: Ronnie Brown
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Subject: categories: Origins of Pursuing Stacks (PS)
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In view of the continuing interest in Alexander's Manuscript with that
title, and the recent articles in Notices AMS on Grothendieck, I have made
a short web page giving giving some information on how that MS came about.
This also explains why PS is in English.
http://www.bangor.ac.uk/~mas010/pursstacks.html
Ronnie Brown
http://www.bangor.ac.uk/~mas010
By the way, the following is the new URL of the updated CPM site
http://www.popmath.org.uk
though the old one in informatics works. Anyone linked to part of the tree
rooted at
http://www.bangor.ac.uk/cpm/
should update.
22Nov2004 16:52:19 0400,2600;00000000000000000000
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Date: Mon, 22 Nov 2004 09:49:41 +0100 (CET)
From: Martin Hofmann
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To: categories@mta.ca
Subject: categories: PhD position in Munich
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Reminder: Call for PhD Applications, deadline 30 Nov 2004

A PhD scholarship is available within the research group for
Theoretical Informatics (Prof Martin Hofmann) at the University of
Munich (LMU).
The successful applicant should have some experience, e.g. in the form
of a topical diploma or master thesis, in one of the following areas:
 Computeraided theorem proving: program logics, decision procedures,
integration of interactive and automatic theorem proving, higherorder
syntax.
 Computational complexity and programming: characterisation of
complexity classes by logical means or programming formalisms, finite
model theory.
 Program analysis: pointers and higherorder store, automatic cost
analysis.
Students with a background in another area of logic in computer
science are also welcome to apply.
The PhD studentship will be associated with the Graduate School
(Graduiertenkolleg) "Logic in Computer Science". The grant is
initially for 36months and will be EUR1000 per month or more. The
successful applicant should participate in undergraduate tuition to
the extent of 2 contact hours per week during term time. Exceptions to
this are negotiable.
Applications comprising
 CV
 record of undergraduate studies
 copy of or link to diploma thesis (perhaps still in draft stage)
 addresses of two senior academics willing to provide references
 1 page essay on why you would like to pursue PhD studies and which
topic / area you would like to work on
are invited until 30th November 2004 and should be sent to
Prof Martin Hofmann
Institut fuer Informatik
Oettingenstr 67
80538 Muenchen
Germany
www.tcs.ifi.lmu.de/~mhofmann
or by email to "hofmann at ifi dot lmu dot de".
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From: "Al Vilcius"
Subject: categories: not vs. non
To: categories@mta.ca
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Has anyone worked out details of relationships between coHeyting =93non=94=
(via
left adjoint to join) and Heyting =93not=94 (via right adjoint to meet), as
these appear in Bill Lawvere's Como 1990 paper =93Intrinsic CoHeyting
Boundaries and the Leibniz Rule in Certain Toposes=94 (SLNM 1488), perhaps =
in
a biHeyting framework and/or in the presence of quantifiers?
Any references would be appreciated.
Thank you kindly.
Al Vilcius
Campbellville, ON
22Nov2004 16:52:19 0400,4053;00000000000000000000
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MessageId: <200411191650.iAJGo9Wr004035@coraki.Stanford.EDU>
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To: categories@mta.ca
Subject: categories: Re: Three papers on selfsimilarity
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Date: Fri, 19 Nov 2004 08:50:08 0800
From: Vaughan Pratt
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Apropos of Tom Leinster's attribution to Peter Freyd of the selfsimilar
construction of the continuum, I hope people won't consider it inappropriate
to repost the first paragraph of Peter's original posting to this list
about his functor.
>Date: Wed, 22 Dec 1999 11:49:50 0500 (EST)
>From: Peter Freyd
>To: categories@mta.ca
>Subject: categories: Real coalgebra
>
>I've been looking at the Proceedings of the Second Workshop on
>Coalgebraic Methods in Computer Science (CMCS'99), Electronic Notes in
>Theoretical Computer Science, Volume 19, to be found at
>
> www.elsevier.nl:80/cas/tree/store/tcs/free/noncas/pc/menu.htm
>
>There's a nice paper by Dusko Pavlovic and Vaughan Pratt. It's
>entitled On Coalgebra of Real Numbers and it has turned me on. Their
>abstract begins:
>
> We define the continuum up to order isomorphism (and hence
> homeomorphism) as the final coalgebra of the functor X x omega,
> ordinal product with omega. This makes an attractive analogy with
> the definition of the ordinal omega itself as the initial algebra of
> the functor 1;X, prepend unity, with both definitions made in the
> category of posets.
>
>I thought of using another functor. And damned if it isn't just what
>I should have had for my CTCS talk last September at Edinburgh.
Selfsimilarity is a common feature of both functors, indeed it is intrinsic
to the construction of the continuum as a final coalgebra. As our functor
makes clear, we constructed the continuum as omega many copies of itself
abutted without overlap, based on the idea that the ordinal omega is in a
sense closed below but open above, a sufficient condition on intervals of
reals to abut to form an interval of reals.
Peter's very nice contribution was to realize that by allowing the copies
to overlap, two copies sufficed. We slapped ourselves on the forehead for
not thinking of that.
Our nonoverlapping selfsimilar construction is treated in
@InProceedings(
PP99, Author="D.~Pavlovi\'c and V.~Pratt",
Title="On coalgebra of real numbers",
BookTitle="Proc. Coalgebraic Methods in Computer Science",
Series="Electronic Notes in Theoretical Computer Science",
Volume="19", Address="Amsterdam", Pages="133147", Year=1999)
The longer journal version appeared as
@Article(
PP02, Author="D.~Pavlovi\'c and V.~Pratt",
Title="The continuuum as a final coalgebra",
Journal=TCS, Volume=280, Number="12", Pages="105122",
Year=2002)
Vaughan Pratt
PS. I hope people aren't strongly opposed to inventors sticking up for
themselves. In the case of the original modal mucalculus, described in
"A Decidable MuCalculus," FOCS'81, 442447, I felt that the judgment of
originality should be left to others, it being selfserving for the inventor
to so judge. The outcome persuaded me that the alternatives for an author in
such a situation were to speak up promptly or run the risk of being forgotten
indefinitely (but hopefully not permanently since history constantly rewrites
itself to reflect the research, interests, and whims of the historians).
De Morgan made a similar judgment call in his celebrated published debate
with the other William Hamilton, not only setting the record straight but
inspiring Boole to write his even more celebrated 1847 pamphlet introducing
Boolean logic.
23Nov2004 20:40:13 0400,27131;00000000000000000000
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Subject: ncategories: foundations and applications
To: categories@mta.ca (categories)
Date: Mon, 22 Nov 2004 21:49:18 0800 (PST)
From: "John Baez"
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I thought some of you might like this....
.......................................................................
Also available at http://math.ucr.edu/home/baez/week209.html
November 21, 2004
This Week's Finds in Mathematical Physics  Week 209
John Baez
Time flies! This June, Peter May and I organized a workshop on
ncategories at the Institute for Mathematics and its Applications:
1) nCategories: Foundations and Applications,
http://www.ima.umn.edu/categories/
I've been meaning to write about it ever since, but I keep
putting it off because it would be so much work. The meeting lasted
almost two weeks. It was an intense, exhausting affair packed
with talks, conversations, and "Russianstyle seminars" where the
audience interrupted the speakers with lots of questions. I took
about 50 pages of notes. How am I supposed to describe all that?!
Oh well... I'll just dive in. I'll quickly list all the official
talks in this conference. I won't describe the many interesting
"impromptu talks", some of which you can see on the above webpage.
Nor will I explain what ncategories are, or what they're good for!
If you want to learn what they're good for, you should go back to
"week73" and read "The Tale of nCategories". And if you want to know
what they *are*, try this brandnew book:
2) Eugenia Cheng and Aaron Lauda, HigherDimensional Categories:
an Illustrated Guide Book, available free online at:
http://www.dpmms.cam.ac.uk/~elgc2/guidebook/
Eugenia and Aaron wrote it specially for the workshop! It's
packed with pictures and it's lots of fun.
I'm just going to list the talks....
Throwing etiquette to the winds, I kicked off the conference
myself with two talks explaining some reasons why ncategories are
interesting and what they should be like:
3) John Baez, Why nCategories? and What ncategories should be like.
Notes available at http://www.ima.umn.edu/categories/#mon
If you're a longtime reader of This Week's Finds you'll know
what I said: ncategories give a new world of math in which equations
are always replaced by isomorphisms, and this world is incredibly rich
in structure. The ncategories called "ngroupoids" magically know
everything there is to know about homotopy theory, while those called
"ncategories with duals" know everything there is to know about the
topology of manifolds. There are, unfortunately, some details that
still need to be worked out!
After my talks there was a reception. Later, over dinner,
Tom Leinster gave a "Russian style seminar" outlining the
different approaches to ncategories:
4) Tom Leinster, Survey and Taxonomy. Talk based on chapter 10
of his book Higher Operads, Higher Categories, Cambridge U. Press,
Cambridge, 2004, also available free online at: math.CT/0305049.
You'll notice these young ncategory people are smart: they
force their publishers to keep their books available for free online!
All scientists should do this, since the only ones who make serious
money from scientific monographs are the publishers. What scientists
get from writing technical books is not money but attention. As
George Franck said, "Attention is a mode of payment... reputation is the
asset into which the attention received from colleagues crystallizes."
The next morning began with a tripleheader talk on "weak categories":
5) Andre Joyal, Peter May and Timothy Porter, Weak categories.
Notes available at http://www.ima.umn.edu/categories/#tues
Here a "weak category" means a category where the usual laws hold
only up to homotopy, where the homotopies satisfy laws of their own
up to homotopy, ad infinitum. If you know what weak infinitycategories
are, you can define a weak category to be one of these where all the
jmorphisms are equivalences for j > 1. But, the nice thing is that
there are ways to define weak categories without the full machinery
of infinitycategories! People have come up with different approaches:
"categories enriched over simplicial sets", "Segal categories",
"A_infinity categories" and also Joyal's "quasicategories". The talk
was a nice introduction to all these approaches.
Then Michael Batanin explained his definition of infinitycategories.
This was a blackboard talk, so there are no notes on the web, but you
can try his original paper:
6) Michael Batanin, Monoidal globular categories as natural
environment for the theory of weak ncategories, Adv. Math. 136
(1998), 39103, also available at
http://www.ics.mq.edu.au/~mbatanin/papers.html
and when you get stuck, try the books by ChengLauda and Leinster.
Over dinner, Eugenia Cheng and Tom Leinster explained the concepts of
"operad" and "multicategory" which play such an important role in so
much work on ncategories. Again there are no notes, so try their books.
I forget when it happened, but sometime around the second or third day
of the conference people decided it was too much of a nuisance listening
to math lectures while eating dinner  mainly because there wasn't enough
room in the dining hall to take notes, and the blackboards weren't big
enough. So at that point, we switched to having lectures *after* dinner.
As I said, this workshop was not for wimps!
The morning of the third day began with a noholdsbarred minicourse
on model categories by Peter May:
7) Peter May, Model categories. Notes available at
http://www.ima.umn.edu/categories/#wed
Model categories are a wonderful framework for relating different
approaches to homotopy theory, and a bunch of people hope they can also
be used to relate different approaches to ncategories.
Then Clemens Berger explained Andre Joyal's approach to weak ncategories:
8) Clemens Berger, Cellular definitions.
Notes available at http://www.ima.umn.edu/categories/#wed
Then, either during or after dinner, Eugenia Cheng explained various
"opetopic" approaches to weak ncategories. Again, the best way to learn
about these is to read the book she wrote with Lauda, or else the book by
Leinster.
On the morning of the fourth day, Andre Joyal explained his work on
quasicategories  an approach to weak categories in which they are
simplicial sets satisfying a restricted version of the Kan condition.
They've been around a long time, but Joyal is redoing all of category
theory in this context! He's been writing a book about this, which
deserves to be called "Quasicategories for the Working Mathematician".
Since Joyal is a perfectionist, this will take forever to finish.
However, we're hoping to extract a preliminary version from him for
the proceedings of this conference. For now, you can read a bit about
quasicategories in Tim Porter's notes mentioned in item 5) above.
Then Tom Leinster and Nick Gurski spoke about Ross Street's definition
to weak infinitycategories, where they are simplicial sets satisfying
an even more subtly restricted version of the Kan condition.
9) Nick Gurski and Tom Leinster, Simplicial definition.
Notes available at http://www.ima.umn.edu/categories/#thur
Street's definition is tough to understand at first, but it should
eventually include Joyal's quasicategories as a special case, which is
nice. For Street's own discussion, see:
10) Ross Street, Weak omegacategories, in Diagrammatic Morphisms
and Applications, eds. David Radford, Fernando Souza, and David Yetter,
Contemp. Math. 318, AMS, Providence, Rhode Island, 2003, pp. 207213.
Also available as www.maths.mq.edu.au/~street/Womcats.pdf
It relies on some work by Dominic Verity which has finally been
written up after many years of unpublished limbo:
11) Dominic Verity, Complicial sets, available as math.CT/0410412.
After dinner we took a turn towards applications, and Larry Breen
explained his work on nstacks and ngerbes. An nstack is like a
sheaf that has an (n1)category of sections, while an ngerbe has an
(n1)groupoid of sections. Such things show up a lot in algebraic
geometry, and more recently in mathematical physics inspired by string
theory. Alas, the audience was rather tired this evening, so Larry
only got to 1stacks and 1gerbes! But he gave an impromptu talk later
where he reached n = 2, and the notes for both talks are available in
combined form here:
12) Larry Breen, nStacks and ngerbes: homotopy theory.
Notes available at http://www.ima.umn.edu/categories/#thur
You've heard about David Corfield's quest for a philosophy of real
mathematics in "week198". He's one of the few philosophers who
understands enough math to realize how cool ncategories are  which
may explain why he's having trouble getting a job. On the morning of
the fourth day, he gave a talk on the impact ncategories could have
in philosophy:
13) David Corfield, nCategory theory as a catalyst for change in
philosophy. Notes available at http://www.ima.umn.edu/categories/#fri
Later that day, Bertrand Toen explained Segal categories, which are
another popular approach to weak categories:
14) Bertrand Toen, Segal categories.
Notes by Joachim Kock available at http://www.ima.umn.edu/categories/#fri
After dinner, he spoke about nstacks and ngerbes:
15) Bertrand Toen, nStacks and ngerbes: algebraic geometry.
Notes by Joachim Kock available at http://www.ima.umn.edu/categories/#fri
Everyone slept all weekend long. Then on Monday of the second week,
the homotopy theorist Zbigniew Fiedorowicz spoke about his work on a
kind of nfold monoidal category that has an nfold loop space as its
nerve. He has some good papers on the web about this, too:
16) Zbigniew Fiedorowicz, nFold categories.
Notes available at http://www.ima.umn.edu/categories/#mon2
C. Balteanu, Z. Fiedorowicz, R. Schwaenzl and R. Vogt,
Iterated monoidal categories, available at math.AT/9808082.
Z. Fiedorowicz, Constructions of E_n operads, available at math.AT/9808089.
Stefan Forcey continued this theme by discussing enrichment over
nfold monoidal categories. He also has a number of papers about
this on the arXiv, of which I'll just mention one:
17) Stefan Forcey, Higher enrichment: nfold Operads and enriched
ncategories, delooping and weakening.
Notes available at http://www.ima.umn.edu/categories/#mon2
Stefan Forcey, Enrichment over iterated monoidal categories,
Algebraic and Geometric Topology, 4 (2004), 95119, available online
at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt47.abs.html
Also available as math.CT/0403152.
After dinner we discussed how to relate different definitions of weak
ncategory.
On Tuesday of the second week, the logician Michael Makkai presented his
astounding project of redoing logic in a way that completely eliminates
the concept of "equality". This *forces* you to do all of mathematics
using weak infinitycategories. I thought this stuff was great, in part
because I finally understood it, and in part because it leads naturally
to the "opetopic" definition of ncategories that James Dolan and I
introduced. The idea of eliminating equality was very much on our mind
in inventing this definition, but we didn't create a system of logic that
systematizes this idea.
There are no notes for Makkai's talk online, but you can get a lot of good
stuff from his website, including:
18) Michael Makkai, On comparing definitions of weak ncategory,
available at http://www.math.mcgill.ca/makkai/
and this more technical paper which works out the details of his vision:
19) Michael Makkai, The multitopic omegacategory of all multitopic
omegacategories, available at http://www.math.mcgill.ca/makkai/
After Makkai's talk, Mark Weber spoke on ncategorical generalizations
of the concept of "monad", which is a nice way of describing mathematical
gadgets. There are no notes for this talk, but his work on higher operads
is at least morally related:
20) Mark Weber, Operads within monoidal pseudo algebras, available as
math.CT/0410230.
Again, after dinner we talked about how to relate different definitions
of weak ncategory.
On Wednesday of the second week, Michael Batanin spoke about his
recent work relating ncategories to nfold loop spaces. Again no
notes, but you can read these papers:
21) Michael Batanin, The EckmannHilton argument, higher operads and
E_nspaces, available at http://www.ics.mq.edu.au/~mbatanin/papers.html
Michael Batanin, The combinatorics of iterated loop spaces,
available at http://www.ics.mq.edu.au/~mbatanin/papers.html
Then Joachim Kock laid the ground for a discussion of ncategories
and topological quantum field theories, or "TQFTs", by explaining the
definition of a TQFT and the classification of 2d TQFTs:
22) Joachim Kock, Topological quantum field theory primer.
Notes available at http://www.ima.umn.edu/categories/#wed2
In the evening, Marco Mackaay and I said more about the relation
between TQFTs and ncategories:
23) Marco Mackaay, Topological quantum field theories.
Notes available at http://www.ima.umn.edu/categories/#wed2
24) John Baez, Space and state, spacetime and process.
Notes available at http://www.ima.umn.edu/categories/#wed2
On Thursday, Ross Street started the day in a pleasantly different
way  he gave a historical account of work on categories and
ncategories in Australia! Australia is home to much of the best
work on these subjects, so if you can understand his history you'll
wind up understanding these subjects pretty well:
25) Ross Street, An Australian conspectus of higher category theory.
Notes available at http://www.ima.umn.edu/categories/#thur2
As a younger exponent of the Australian tradition, it was then nicely
appropriate for Steve Lack to speak about ways of building a model
category of 2categories:
26) Steve Lack, Higher model categories. Notes available at
http://www.ima.umn.edu/categories/#thur2
In the afternoon we had a blast of computer science. First John Power
gave a hilarious talk phrased in terms of how one should convince
computer theorists to embrace categories, then 2categories, and then
maybe higher categories:
27) John Power, Why tricategories? Notes available at
http://www.ima.umn.edu/categories/#thur2
I spoke about Power's paper with this title back in "week53"; now
you can get it online!
Then Phillipe Gaucher, Lisbeth Falstrup and Eric Goubault spoke about
higherdimensional automata and directed homotopy theory:
28) Phillipe Gaucher, Towards a homotopy theory of higher dimensional
automata. Notes available at http://www.ima.umn.edu/categories/#thur2
Lisbeth Falstrup, More on directed topology and concurrency,
Notes available at http://www.ima.umn.edu/categories/#thur2
Eric Goubault, Directed homotopy theory and higherdimensional automata,
Notes available at http://www.ima.umn.edu/categories/#thur2
On Friday, Martin Hyland and Tony Elmendorf gave a doubleheader
talk on higherdimensional linear algebra and how some concepts in
this subject can be simplified using symmetric multicategories.
There are, alas, no notes for this talk. You just had to be there.
Finally, my student Alissa Crans gave a talk on higherdimensional
linear algebra, with an emphasis on categorified Lie algebras:
29) Alissa Crans, Higher linear algebra. Notes available at
Notes available at http://www.ima.umn.edu/categories/#fri2
Hers was the last talk in the workshop! I would like to say more about
it, but I'm exhausted... and her talk fits naturally into a discussion of
"higher gauge theory", which deserves a Week of its own.
By the way, you can see pictures of this workshop here:
30) John Baez, IMA, http://math.ucr.edu/home/baez/IMA/
If you want to see what these crazy ncategory people look like,
you can see most of them here.
Hmm. If you wanted me to actually *explain* something this week, I'm
afraid you'll be rather disappointed  so far everything has just
been pointers to other material.
Luckily, while I was at this workshop I wrote a little explanation
of some material on Picard groups and Brauer groups. There's a
Spanish school of higherdimensional algebra, centered in Granada, and
this spring Aurora del Rio Cabeza came from Granada to visit UCR.
She and James Dolan spent a lot of time talking about categorical
groups (also known as "2groups") and cohomology theory. I was, alas,
too busy to keep up with their conversations, but I learned a little
from listening in... and here's my writeup!
Higher categories show up quite naturally in the study of
commutative rings and associative algebras over commutative rings.
I'd heard of things called "Brauer groups" and "Picard groups"
of rings, and something called "Morita equivalence", but I only
understood how these fit together when I learned they were part
of a marvelous thing: a weak 3groupoid!
Here's how it goes. You don't need to know much about higher
categories for this to make some sense... at least, I hope not.
Starting with a commutative ring R, we can form a weak 2category
Alg(R) where:
an object A is an associative algebra over R
a 1morphism M: A > B is an (A,B)bimodule
a 2morphism f: M > N is a homomorphism between (A,B)bimodules.
This has all the structure you need to get a 2category. In particular,
we can "compose" an (A,B)bimodule and a (B,C)bimodule by tensoring them
over B, getting an (A,C) bimodule. But since tensor products are only
associative up to isomorphism, we only get a *weak* 2category, not a
strict one.
This weak 2category has a tensor product, since we can tensor two
associative algebras over R and get another one. All the stuff listed
above gets along with this process! When an ncategory has a wellbehaved
tensor product we call it "monoidal", so Alg(R) is a weak monoidal
2category. But using a standard trick we can reinterpret this as a weak
3category with one object, as follows:
there's only one object, R
a 1morphism A: R > R is an associative algebra over R
a 2morphism M: A > B is an (A,B)bimodule
a 3morphism f: M > N is a homomorphism between (A,B)bimodules.
Note how all the morphisms have shifted up a notch. What used to be
called objects, the associative algebras over R, are now called
1morphisms. We "compose" them by tensoring them over R.
Next, recall a bit of ncategory theory from "week35". In an ncategory
we define a jmorphism to be an "equivalence" iff it's invertible... up
to equivalence! This definition may sound circular, but really just
recursive. To start it off we just need to add that an nmorphism is
an equivalence iff it's invertible.
What does equivalence amount to in the 3category Alg(R)? It's easiest
to figure this out from the top down:
A 3morphism f: M > N is an equivalence iff it's invertible, so it's
an isomorphism between (A,B)bimodules.
A 2morphism M: A > B is an equivalence iff it's invertible up to
isomorphism, meaning there exists N: B > A such that:
M tensor_B N is isomorphic to A as an (A,A)bimodule,
N tensor_A M is isomorphic to B as a (B,B)bimodule.
In this situation people say M is a "Morita equivalence" from A to B.
A 1morphism A: R > R is an equivalence iff it's invertible up to
Morita equivalence, meaning there exists a 1morphism B: x > x
such that:
A tensor_R B is Morita equivalent to R as an associative algebra over R,
B tensor_R A is Morita equivalent to R as an associative algebra over R.
In this situation people say A is an "Azumaya algebra".
Here's a nice example of how Morita equivalence works. Over any commutative
ring R there's an algebra R[n] consisting of n x n matrices with entries
in R. R[n] isn't usually isomorphic to R[m], but they're always Morita
equivalent! To see this, suppose
M: R[n] > R[m] is the space of n x m matrices with entries in R,
N: R[m] > R[n] is the space of m x n matrices with entries in R.
These become bimodules in an obvious way via matrix multiplication, and
a little calculation shows that they're inverses up to isomorphism!
So, all the algebras R[n] are Morita equivalent. In particular this
means that they're all Morita equivalent to R, so they are Azumaya
algebras of a rather trivial sort.
If we take R to be real numbers there is also a more interesting
Azumaya algebra over R, namely the quaternions H. This follows from
the fact that
H tensor_R H = R[4]
This says H tensor_R H is Morita equivalent to R as an associative
algebra over R, which implies (by the definition above) that H is an
Azumaya algebra.
Morita equivalence is really important in the theory of C*algebras,
Clifford algebras, and things like that. Someday I want to explain
how it's connected to Bott periodicity. Oh, there's so much I want
to explain....
But right now I want to take our 3category Alg(R), massage it a bit,
and turn it into a topological space! Then I'll look at the homotopy
groups of this space and see what they have to say about our ring R.
To do this, we need a bit more ncategory theory. A weak ncategory
where all the 1morphisms, 2morphisms and so on are equivalences is
called a "ngroupoid". For example, given any weak ncategory, we can
form a weak ngroupoid called its "core" by throwing out all the
morphisms that aren't equivalences.
So, let's take the core of Alg(R) and get a weak 3groupoid. Here's
what it's like:
there's one object, R
the 1morphisms A: x > x are Azumaya algebras over R
the 2morphisms M: A > B are Morita equivalences
the 3morphisms f: M > N are bimodule isomorphisms.
Next, given a weak ngroupoid with one object, it's very nice to compute
its "homotopy groups". These are easy to define in general, but I'll
just do it for the core of Alg(R) and let you guess the general pattern.
First, notice that:
the identity 1morphism 1_R: R > R is just R, regarded as an associative
algebra over itself in the obvious way
the identity 2morphism 1_{1_R}: 1_R > 1_R is just R, regarded as an
(R,R)bimodule in the obvious way
the identity 3morphism 1_{1_{1_R}}: 1_{1_R} > 1_{1_R} is just the
identity function on R, regarded as an isomorphism of (R,R)bimodules.
At this point we let out a cackle of ncategorical glee. Then,
we define the homotopy groups of the core of Alg(R) as follows:
the 1st homotopy group consists of equivalence classes of
1morphisms from R to itself
the 2nd homotopy group consists of equivalence classes of
2morphisms from 1_R to itself
the 3rd homotopy group consists of equivalence classes of
3morphisms from 1_{1_R} to itself
Here we say two morphisms in an ncategory are "equivalent" if there is
an equivalence from one to the other (or if they're equal, in the case
of nmorphisms).
I hope the pattern in this definition of homotopy groups is obvious.
In fact, ngroupoids are secretly "the same"  in a subtle sense I'd
rather not explain  as spaces whose homotopy groups vanish above
dimension n. Using this, the homotopy groups as defined above turn
out to be same as the homotopy groups of a certain space associated
with the ring R! So, we're doing something very funny: we're using
algebraic topology to study algebra.
But, we don't need to know this to figure out what these homotopy
groups are like. Unraveling the definitions a bit, one sees they
amount to this:
The 1st homotopy group consists of Morita equivalence classes of
Azumaya algebras over R. This is also called the BRAUER GROUP of R.
The 2nd homotopy group consists of isomorphism classes of Morita
equivalences from R to R. This is also called the PICARD GROUP of R.
The 3rd homotopy group consists of invertible elements of R. This is
also called the UNIT GROUP of R.
People had been quite happily studying these groups for a long time
without knowing they were the homotopy groups of the core of a weak
3category associated to the commutative ring R! But, the relationships
between these groups are easier to explain if you use the ncategorical
picture. It's a great example of how ncategories unify mathematics.
For example, everything we've done is functorial. So, if you have a
homomorphism between commuative rings, say
f: R > S
then you get a weak 3functor
Alg(f): Alg(R) > Alg(S)
This gives a weak 3functor from the core of Alg(R) to the core of Alg(S),
and thus a map between spaces... which in turn gives a long exact sequence
of homotopy groups! So, we get interesting maps going from the unit,
Picard and groups of R to those of S  and these fit into an interesting
long exact sequence.
For more, try the following papers. The first paper is actually about a
generalization of Azumaya algebras called "Azumaya categories", but it
starts with a nice quick review of Azumaya algebras and Brauer groups:
31) Francis Borceux and Enrico Vitale, Azumaya categories,
available at http://www.math.ucl.ac.be/AGEL/Azumaya_categories.pdf
Category theorists will enjoy the generalization: since algebras are
just oneobject categories enriched over Vect, the concept of Azumaya
algebra really *wants* to generalize to that of an Azumaya category.
I'm sure most of the BrauerPicardMorita stuff generalizes too, but I
haven't checked that out yet.
This second paper makes the connection between Picard and Brauer
groups explicit using categorical groups:
32) Enrico Vitale, A PicardBrauer exact sequence of categorical groups,
Journal of Pure and Applied Algebra 175 (2002) 383408.
Also available as http://www.math.ucl.ac.be/membres/vitale/catgruppi2.pdf

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
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Date: Tue, 23 Nov 2004 06:29:55 0500 (EST)
From: Peter Freyd
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Subject: categories: on Vaughn and Tom
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Vaughn's point is a good one, but not quite apropos of Tom's notion of
selfsimilarity. In the category of ordered sets the final coalgebra
of the PavlovicPratt functor is, indeed, the halfopen real interval,
[0,1). But Tom was in the category of topological spaces. To make [0,1)
into a selfsimilar _space_ one must reinterpret their functor so that
the new point of the onepoint compactification of each copy of X is
identified with the bottom of the next copy.
So what interesting space was first described as the final coalgebra
of a functor? (I think that locution succeeds in avoiding such things
as constant functors.) I would suggest the onepoint compactification
of the countable discrete set. (The functor, of course, is 1 + X.)
Peter
24Nov2004 20:03:48 0400,2672;00000000000000000000
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From: Peter Selinger
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Subject: categories: MSCS special issue on quantum programming languages
To: categories@mta.ca (Categories List)
Date: Tue, 23 Nov 2004 22:07:15 0500 (EST)
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CALL FOR PAPERS
Mathematical Structures in Computer Science
Special Issue on
Quantum Programming Languages
The 2nd International Workshop on Quantum Programming Languages was
held July 1213, 2004 in Turku, Finland as an affiliated LICS
workshop. As previously announced, there will be a special issue of
the journal Mathematical Structures in Computer Science (MSCS,
Cambridge University Press) devoted to areas represented at the
workshop. Submissions for this special issue are hereby solicited.
Topics. The topics of the workshop, and of the special issue, include
the following. Syntax and semantics of quantum programming languages,
new paradigms for quantum programming, specification of quantum
algorithms, higherorder quantum computation, quantum data types,
reversible computation, axiomatic approaches to quantum computation,
concurrent and distributed quantum computation, compilation of quantum
programs, semantical methods in quantum information theory, and
categorical models for quantum computation.
Submission procedure. Submissions should be sent electronically to
Peter Selinger by January 25, 2005. To
prevent email problems, successful receipt of each submission will be
acknowledged by return email. There are no special formatting
requirements (e.g. page limits) beyond the general editorial policy of
MSCS. However, authors are asked to take into consideration that there
is a **finite bound on the total number of pages** in the special
issue, and therefore, the ratio of content+quality / number of pages
is expected to be very high. Submission is not restricted to workshop
participants. Submissions will be refereed according to the usual
very high standards of MSCS. Peter Selinger will serve as guest editor
for this special issue. Final decisions on editorial matters rest with
the editorinchief of MSCS.
24Nov2004 20:03:48 0400,1194;00000000000000000000
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Subject: categories: Re: on Vaughan and Tom
From: "Tom Leinster"
To:
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> So what interesting space was first described as the final coalgebra of
> a functor? [...] I would suggest the onepoint compactification of
> the countable discrete set. (The functor, of course, is 1 + X.)
How about the Cantor set (as the final coalgebra for the endofunctor of
Top defined by X > X + X )?
Tom
24Nov2004 20:03:48 0400,1045;00000000000000000000
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Date: Wed, 24 Nov 2004 12:20:16 0500
From: Galchin Vasili
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To: categories@mta.ca
Subject: categories: D. E. Rydeheard's computational category theory project
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Hello Cat Community,
Recently I stumbled across a softcopy of Rydeheard's book
"Computational Category Theory" at
http://www.cs.man.ac.uk/~david/categories/index.html
I would like to know of other ongoing projects on computational
category theory) and possibly how they differ from Rydeheard's
approach.
Regards, Bill Halchin
24Nov2004 20:03:48 0400,2593;00000000000000000000
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Date: Tue, 23 Nov 2004 20:07:50 0800
From: Dusko Pavlovic
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Peter Freyd wrote:
>Vaughn's point is a good one, but not quite apropos of Tom's notion of
>selfsimilarity. In the category of ordered sets the final coalgebra
>of the PavlovicPratt functor is, indeed, the halfopen real interval,
>[0,1). But Tom was in the category of topological spaces. To make [0,1)
>into a selfsimilar _space_ one must reinterpret their functor so that
>the new point of the onepoint compactification of each copy of X is
>identified with the bottom of the next copy.
the title of tom's series of papers suggests that he wants to develop a
General Theory of SelfSimilarity, not just a topological theory. and
in a general theory, the various decompositions of [0,1) into the
copies of [0,1) and so on  would qualify as selfsimilarity in
most people's eyes.
i didn't read tom's papers yet, but it seems to me that the essence of
selfsimilarity in general, and the conceptual value of all of our
reconstructions of the reals using selfsimilarity, is not so much in
their topology or order, but in their coalgebra, capturing the infinite
subdivisions.
and finally, to pop up a level, i am a bit surprised with tom's line of
reasoning: "it's not topology, so it shouldn't be mentioned", tacitly
supported by peter's posting. i was under the impression that the authors
in all sciences consider it to be a matter of good taste to be generous
in references and acknowledgements, and not frugal; to be inclusive and
not exclusive. people provide as many references as they can, to let the
readers track their ideas, and make the connections. useful papers don't
cite just their parent papers, but also their grandparent papers, and
brothers and sisters. especially when the parents so clearly and
generously recognize their debts, as peter's original postings did.
 dusko
25Nov2004 14:18:16 0400,1422;00000000000000000000
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Date: Thu, 25 Nov 2004 11:35:53 +0100
From: Alexey Cherchago
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Dear Cat Community,
I have the following question about preservation of properties of fiber
categories in the total category:
Given a contravariant indexed category F : B^Op > Cat, and the
Grothendieck translation from the indexed category to fibrations
Flat(F). Then, the first projection P : Flat(F) > B forms a (split)
fibration, called the flattening of F.
Now assume that the fiber categories are complete/cocomplete/adhesive.
Would it be the case that the total category Flat(F) of the described
(split) fibration is also complete/cocomplete/adhesive?
Any references would be appreciated.
Thank you kindly.
Best regards,
Alexey Cherchago
25Nov2004 14:18:16 0400,2139;00000000000000000000
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Date: Wed, 24 Nov 2004 20:09:16 0500
From: jim stasheff
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Dusko Pavlovic wrote:
> i was under the impression that the authors
> in all sciences consider it to be a matter of good taste to be generous
> in references and acknowledgements, and not frugal; to be inclusive and
> not exclusive. people provide as many references as they can, to let the
> readers track their ideas, and make the connections. useful papers don't
> cite just their parent papers, but also their grandparent papers, and
> brothers and sisters. especially when the parents so clearly and
> generously recognize their debts, as peter's original postings did.
>
>  dusko
>
>
>
>
not all authors  unfortunately
though in many cases it may be a lack of knowledge/scholarship
generosity and scholarship in references is to be applauded
but acknowledgements in a certian science can be so profuse
naming everybody the author had a slighly related conversation with
there by diluting the major influences
another extreme in tha same science is to give a bibliogrpahy without titles
this has multiple disadvantages
at one conference to a mixed audience (math and ...) the math organizer
insisted references include title
to which a participant repsonded
what if you don't know the title!!!
jim
>
>
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Date: Thu, 25 Nov 2004 08:54:16 0800
From: Vaughan Pratt
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>> Peter F:
>> So what interesting space was first described as the final coalgebra of
>> a functor? [...] I would suggest the onepoint compactification of
>> the countable discrete set. (The functor, of course, is 1 + X.)
>Tom L:
>How about the Cantor set (as the final coalgebra for the endofunctor of
>Top defined by X > X + X )?
I'd love to know the date Peter has in mind for his candidate. Meanwhile
these are both nice candidates. In the interest (inter alia :) of settling
Peter's question in an orderly fashion, I'd be happy to serve as a clearing
house for dates for spaces first presented as a final coalgebra of a functor
Since Peter's question didn't impose Tom's implicit assumption that the
functor be on Top (so to speak) (and moreover Peter's original example of
a finitary functor for the continuum was not on Top but on Pos+, posets
with top and bottom), I'll accept functors on any category.
Obviously the question has its specializations to people's favorite
categories, I'll handle just the general list and let others worry about
the preferredcategory lists.
I can prime the pump with the candidates I'm aware of, ordered by date of
description as a final coalgebra. Contributions can take the form either
of an earlier publication date for a candidate on the list (with reference
of course), or a space not yet on the list.
CANDIDATE DATE REFERENCE
Baire space Mar 99 Pavlovic&Pratt, CMCS'99
The continuum Mar 99 Pavlovic&Pratt, CMCS'99
Cantor space Mar 99 Pavlovic&Pratt, CMCS'99
Wilson space Dec 99 Peter's categories posting, 12/23/99
1pt comp'n of discrete N Nov 04 Peter's categories posting, 11/23/04
The nth item on this list for n<5 can be described as the continuum with
n1 copies of each rational.
As a generalization of Wilson space (which itself can be regarded as an
extension of Cantor space), this list (less the discrete N example) can
be extended further via the finitary functor X;(n3);X where ; is poset
concatenation and n3 denotes the chain with n3 elements, for n >= 3.
Thus Turkey Space (today being Thanksgiving in North America), as the
continuum with each rational in quadruplicate, would be described by the
functor X;2;X, and so on with X;3;X etc. describing spaces that hopefully
will rename nameless.
Vaughan
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Subject: categories: Contexts for selfsimilarity
From: Tom Leinster
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> i am a bit surprised with tom's line of reasoning: "it's not
> topology, so it shouldn't be mentioned"
I don't know what I wrote suggesting that I think that; I don't.
In two of the papers advertised (first paragraph of the first
paper and bottom of p.3 of the second), I listed some other types
of selfsimilarity that I'd like to investigate. Let me expand
on the possibilities.
1, 2. Settheoretic and topological: these are the types of
selfsimilarity that I understand by far the best at
present.
3. Typetheoretic: recursive datatypes can be understood as
selfsimilar objects, the bestknown example being trees
(which can be characterized as a final coalgebra). For
instance, given a polynomial p(x) in N[x], you can consider
solutions to x = p(x) (in rings, rigs, distributive
categories, and rig categories): see the work of Robbie Gates,
Peter Hines (TAC vol 6), Marcelo Fiore and I, and probably
many others.
4. Conformal/analytic: this is the natural setting for discussing
the selfsimilarity of Julia sets of complex rational
functions.
5. Metric: e.g. the Koch snowflake is topologically just a
circle, but has interesting metric selfsimilarity.
6. Measuretheoretic: cf. some of Peter's postings of 19992000
concerning integration.
7. Ordertheoretic: both Peter's and Dusko's/Vaughan's
characterizations of a real interval produce its ordering. (I
have some idea of how to handle order in the much more general
situation discussed in my papers, but it's early days.)
8. Categorical: e.g. the category of strict omegacategories is
the terminal coalgebra for the endofunctor of CAT defined by V
> VCat. (This was an idea of Carlos Simpson.) The same
goes for globular sets (omegagraphs), changing VCat to
VGraph.
9. Statistical: there are socalled random fractals  e.g. take a
black square; divide it into a 3x3 grid and white out each of
the 9 subsquares with probability p; do the same to each black
subsquare; continue ad infinitum.
10. Algebraic: the Thompson groups are in some sense highly
selfsimilar. (These groups may be best known to readers of
this list from Freyd and Heller's work on homotopy
idempotents, but are also very treey in nature.)
Here's a question belonging to (10), to which I don't know the
answer. Let C be the category whose objects are triples (V, v_0,
v_1) where V is a vector space and v_0 and v_1 are linearly
independent vectors in V, and whose maps preserve linear structure
and the `basepoints'. There's a `wedge' functor C x C > C
defined by
(V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1)
where (V + W)/~ is the direct sum with v_1 identified with w_0.
(So dim(V wedge W) = dim V + dim W  1.) There's then an
endofunctor G of C given by selfwedging. Question: what, if
any, is the terminal Gcoalgebra?
(I suspect the answer is something to do with measure/integration
 again see Peter's previous postings  but really have no idea.)
Finally, re citations: I'll stick in a PavlovicPratt reference,
as suggested.
All the best,
Tom
26Nov2004 13:34:46 0400,1447;00000000000000000000
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Date: Thu, 25 Nov 2004 15:44:46 0500 (EST)
From: Peter Freyd
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Subject: categories: Wilson space
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Another little point: Vaughan credits me with characterizing Wilson space
as a final coalgebra. Well, not the topological space. What I wrote in
that first posting was:
Just for comparison, consider the category of posets and the functor
that sends X to X;1;X. The open interval is an invariant object
for this functor but it is not the final coalgebra. For that we need
 as we called it in Cats and Alligators  Wilson space. Actually,
not the space but the linearly ordered set, most easily defined as the
lexicographically ordered subset, W, of sequences with values in
{1, 0, 1} consisting of all those sequences such that for all n
a(n) = 0 => a(n+1) = 0 (take a finite word on {1,1} and pad it
out to an infinite sequence by tacking on 0s).
There is, indeed, a functor on Top that delivers Wilson _space_, to wit,
the one that sends X to the scone of X + X.
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Subject: categories: history
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One little point: in my first 1999 posting on the subject I wrote:
In fact, we didn't need to start in the category of posets. It would
have sufficed to work in the category of sets with distinct top and
bottom...The final coalgebra is still the closed interval and, yes,
the ordering is implicit.
But the first explicit final coalgebra characterization of the
closed interval as a _space_ seems to have came a month later:
Date: Mon, 24 Jan 2000 20:14:36 +0100 (MET)
From: "Martin H. Escardo"
To: categories@mta.ca
Subject: categories: Freyd's couniversal characterization of [0,1]
It would be interesting to test Freyd's couniversal characterization
of the unit interval in many other categories.
Here I test it in Top, the category of topological spaces and
continuous maps, and various full subcategories, where one would hope
to get the unit interval with the Euclidean topology.

Summary of the outcome of some tests:
(1) In Top, the final coalgebra for Freyd's functor exists. Its
underlying object, however, is an indiscrete space (unsurprisingly).
(2) In the category of T0 spaces, it doesn't exist.
(3) In the category of normal spaces it does exist, and, as one
would hope, its underlying object is indeed the unit interval with the
Euclidean topology.
SNIP
For the rest see
north.ecc.edu/alsani/ct9900(812)/msg00082.html
There was an important followup:
Date: Thu, 27 Jan 2000 13:25:25 +0100 (MET)
From: "Martin H. Escardo"
To: categories@mta.ca
Subject: categories: re: Freyd's couniversal characterization of [0,1]
The following is of course rather unpleasent:
> (1) In Top, the final coalgebra for Freyd's functor exists. Its
> underlying object, however, is an indiscrete space (unsurprisingly).
This can be fixed by choosing a slightly different category of
bipointed objects.
Define a *regularly bipointed object* to be an object X with two
distinguished points x0,x1:1>X such that [x0,x1]:1+1>X is regular
mono. Then the terminal coalgebra for Freyd's functor is 1 iff 1+1=1.
With the restriction to regularly bipointed topological spaces, the
statement (1) becomes false because the twopoint discrete space 1+1
is not homeomorphically embeded as a subspace of any indiscrete space.
Hopefully, there isn't a final coalgebra in RegBi(Top), but I don't
see any if there is, it cannot be the Euclidean interval.
SNIP
26Nov2004 14:25:47 0400,3203;00000000000000000000
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From: Michael Mislove
Subject: categories: Re: on Vaughan and Tom
Date: Thu, 25 Nov 2004 17:34:53 0600
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This thread reminds me of John Rhodes' review of the classic, "The
Algebraic Theory of Semigroups, Vol. I" by Clifford and Preston. This
exceptional account of the state of semigroup theory in the early 60s
contained more than copious citations, to authors for published works,
to colleagues for ideas, and even to the extent that solutions to
problems were cited. Rhodes, his acerbic wit at hand, commented on this
incredible documentation of where even the simplest ideas had
originated, was led to comment, "it's surprising that the authors
didn't cite Gutenberg for the type."
Mike Mislove
On Nov 24, 2004, at 7:09 PM, jim stasheff wrote:
> Dusko Pavlovic wrote:
>
>> i was under the impression that the authors
>> in all sciences consider it to be a matter of good taste to be
>> generous
>> in references and acknowledgements, and not frugal; to be inclusive
>> and
>> not exclusive. people provide as many references as they can, to let
>> the
>> readers track their ideas, and make the connections. useful papers
>> don't
>> cite just their parent papers, but also their grandparent papers, and
>> brothers and sisters. especially when the parents so clearly and
>> generously recognize their debts, as peter's original postings did.
>>
>>  dusko
>>
>>
>>
>>
>
> not all authors  unfortunately
> though in many cases it may be a lack of knowledge/scholarship
>
> generosity and scholarship in references is to be applauded
> but acknowledgements in a certian science can be so profuse
> naming everybody the author had a slighly related conversation with
> there by diluting the major influences
>
> another extreme in tha same science is to give a bibliogrpahy without
> titles
> this has multiple disadvantages
> at one conference to a mixed audience (math and ...) the math
> organizer
> insisted references include title
> to which a participant repsonded
> what if you don't know the title!!!
>
> jim
>
>
>>
>>
>
>
>
===============================================
Professor Michael Mislove Phone: +1 504 8623441
Department of Mathematics FAX: +1 504 8655063
Tulane University URL: http://www.math.tulane.edu/~mwm
New Orleans, LA 70118 USA
===============================================
26Nov2004 14:25:47 0400,1826;00000000000000000000
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Date: Thu, 25 Nov 2004 16:20:09 0500 (EST)
From: Peter Freyd
MessageId: <200411252120.iAPLK9LD006322@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: categories: Mandelbrot and the selfsimilar
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I've never before looked at Mandelbrot's original paper on self
similar geometric objects. Damned if he doesn't have as his first
formal example the interval. He points out that it the same as the
result of taking N copies, diminishing each by a factor of N, and
then stringing them together.
So when N = 2 he was pointing out that the closed interval was an
invariant object of the functor I used. (No, of course, he didn't
point out that it was a final coalgebra. And he was thought he was
talking about the halfopen interval, not the closed interval.)
He goes on to tell how to make other selfsimilar curves: Choose a
polygonal curve with N straight pieces, all the same length. Then:
"Replace each of its N legs by a curve deduced from the whole.....
through similarity of a fixed ratio. One is left with a curve made of
N^2 legs;.....the desired selfsimilar curve is approaced by an
infinite sequence of these steps."
It can all be found on JSTORE:
How Long Is the Coast of Britain? Statistical SelfSimilarity and
Fractional Dimension
Benoit Mandelbrot
Science, New Series, Vol. 156, No. 3775. (May 5, 1967), pp. 636638.
("Statistical" because the coast of Britain is not selfsimilar in
the strict formal sense.)
26Nov2004 14:26:21 0400,1904;00000000000100000000
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Date: Fri, 26 Nov 2004 08:04:32 +0100
Subject: categories: Re: question about fibrations
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To: Categories
From: jean benabou
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The answer to the question is obviously NO unless one makes much=20
stronger assumptions on F and B.
Take for F the constant functor with value the terminal category 1, the=20=
category denoted by Flat(f) is then (canonically isomorphic to) B,=20
whereas the fibers have all completeness properties one can desire.
Le jeudi, 25 nov 2004, =E0 11:35 Europe/Paris, Alexey Cherchago a =E9crit =
:
> Dear Cat Community,
>
> I have the following question about preservation of properties of =
fiber
> categories in the total category:
>
> Given a contravariant indexed category F : B^Op > Cat, and the
> Grothendieck translation from the indexed category to fibrations
> Flat(F). Then, the first projection P : Flat(F) > B forms a (split)
> fibration, called the flattening of F.
>
> Now assume that the fiber categories are complete/cocomplete/adhesive.
> Would it be the case that the total category Flat(F) of the described
> (split) fibration is also complete/cocomplete/adhesive?
>
> Any references would be appreciated.
> Thank you kindly.
>
> Best regards,
> Alexey Cherchago
>
>
>
26Nov2004 14:26:49 0400,3309;00000000000000000000
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Date: Fri, 26 Nov 2004 14:40:21 +0000
From: Claudio Hermida
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> Dear Cat Community,
>
> I have the following question about preservation of properties of fiber
> categories in the total category:
>
> Given a contravariant indexed category F : B^Op > Cat, and the
> Grothendieck translation from the indexed category to fibrations
> Flat(F). Then, the first projection P : Flat(F) > B forms a (split)
> fibration, called the flattening of F.
>
> Now assume that the fiber categories are complete/cocomplete/adhesive.
> Would it be the case that the total category Flat(F) of the described
> (split) fibration is also complete/cocomplete/adhesive?
>
> Any references would be appreciated.
> Thank you kindly.
>
> Best regards,
> Alexey Cherchago
Given a fibration P: Flat(F) > B, assuming B is complete then
(E is complete and P continuous) iff P is fibred complete (complete
fibres and continuous reindexing) .
Dually for cofbirations (coming from 'covariant' indexed categories) and
cocompleteness.
I am clueless about 'adhesive categories' but Google points out to the
following def:
C adhesive means it has pullbacks, pushout along monos and these latter
satisfy a certain exactness condition involving pullback stability (a
socalled "VK square" which is actually a cube(?)).
Assuming P: Flat(F) > B preserves monos, an arrow in Flat(F) is monic
iff its image in B and its vertical factor are monic. Ergo, assume:
B adhesive
P has direct images along monos (m^* have left adjoints)
Then, (P has fibrewise pb/po along monos) implies (Flat(F) has same and
P preserves that).. The VK condition is a little obtruse, and at first
sight requires the following additional assumption: the ' po along
monos' in Flat(F) preserves cartesian morphisms and reindexing functors
preserve po along monos. Then it seems that one gets (if one cares to do
the relevant calculations)
P fibrewise adhesive iff Flat(P) adhesive
(for the only if, given a stack of 2 cubes where the bottom one has
all side faces pb, the total cube is a VK square iff the top one is such).
References:
On the correspondence of limits/colimits between fibres and total
Flat(F) see:
 Gray, J. W., Fibred and Cofibred categories, Proceedings of the
Conference on Categorical
Algebra,1966.
There are more recent references which treat this subject (fibred vs.
global properties) from a more abstract point of view (and it a broader
context), but they are probably not the first place to look if one is
not acquainted with the basics.
26Nov2004 14:46:11 0400,2911;00000000000100000000
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Date: Wed, 24 Nov 2004 12:09:21 +0100 (CET)
From: Jiri Adamek
To: categories@mta.ca
Subject: categories: Open letter to F.W. Lawvere: Future of Category Theory conferences
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Dear Bill:
As you know, there have been several years recently when there has been no
general conference on Category Theory during the summer. We believe it
would be beneficial for the future of the subject if there were some way
of ensuring that such conferences take place more regularly (say at least
once every two years), and so we should like to propose the establishment
of an international Advisory Committee which would solicit and consider
proposals from potential organizers of such conferences. We think it is
important that individual initiative in organizing conferences should not
be stifled, and so we envisage that the Advisory Committee would be barred
from imposing conditions on the way a particular conference was organized,
although it might of course offer advice to an individual organizer if it
was requested. We also envisage that a number of conferences with
categorical topics would be organized independently of the activities of
the Advisory Committee, and that the successful regional series such as
the Peripathetic, Octoberfest, Midwest, etc. will be organized
independently of the committee. The same applies to meetings on
specialized topics.
We should be very pleased if you would agree to act as Chair of the
Advisory Committee and to choose the other members of the committee soon.
Our proposal of you as a Chair does not mean that we expect
you to do most of the chores: we can imagine a model of rotating
ViceChairs, one for every conference. In addition to a small Advisory
Committee, all the colleagues who have previously organized such
international CT meetings over the past quarter century should be
considered as a larger committee to which younger wouldbe organizers can
appeal for encouragement and concrete advice. The small committee could
facilitate such contacts, thus providing a unified channel of
communication which is however less public than the category
network.
We very much hope that you will agree to undertake this responsibility.
Thank you for your cooperation,
with best regards
Jiri Adamek Bob Rosebrugh
Francis Borceux Jiri Rosicky
Dominique Bourn Ross Street
George Janelidze Walter Tholen
Peter Johnstone
26Nov2004 19:47:23 0400,898;00000000000000000000
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Date: Fri, 26 Nov 2004 20:18:14 +0100
From: Krzysztof Worytkiewicz
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Dear all,
Here is a little note on a Thomasonstyle model structure on 2Cat:
http://arxiv.org/abs/math.AT/0411154
Cheers
Krzysztof
26Nov2004 19:47:23 0400,7796;00000000000000000000
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MessageID: <41A7A4D1.6020107@sce.carleton.ca>
Date: Fri, 26 Nov 2004 16:49:05 0500
From: Lionel Briand
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Subject: categories: The premier conference in ModelDriven Development  2005 edition
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<>Apologies if you receive multiple copies of this CFP.
************************************************************************
Call for Papers (scientific, experience, Workshop, tutorials, etc.)
MoDELS 2005
ACM/IEEE 8th International Conference on Model Driven Engineering
Languages and Systems (formerly the UML series of conferences)
October 2  7, 2005
Half Moon Resort, Montego Bay, JAMAICA
http://www.modelsconference.org/
************************************************************************
<> <> <> and more !!!
The growing complexity of computerbased systems has resulted in an
awareness of the need to raise the level of abstraction at which such
systems are developed. Modeldriven development approaches and
technologies, in which development is centered round the manipulation of
models, address this need.
MoDELS 2005 is the premier conference devoted to the topic of
modeldriven engineering of softwarebased systems, and covers both
languages and development frameworks used to create and evolve complex
software systems. This conference is both an expansion and a
redirection of previous Unified Modeling Language (UML) conferences,
and replaces that series of conferences for 2005 and beyond. While the
UML has played a significant role in the development of software
modeling approaches in both academia and industry, there is a need to
address modeling issues that go beyond the use of the UML. Initiatives
such as the OMG's Model Driven Architecture (MDA) and Model Integrated
Computing (MIC) are a reflection of the desire to shift attention to the
broader problem of providing costeffective support for modeldriven
development. The MoDELS series of conferences is intended to be the
premier venue for the exchange of innovative technical ideas and
experience related to modeldriven development of softwarebased systems.
MoDELS 2005 will include both scientific and experience conference
papers, workshops, tutorials, a doctoral symposium, posters and a tool
challenge. (See conference web site for further details)
Scientific papers: We invite scientific research papers describing
innovative research on modeldriven development, and innovative research
on other aspects of modeling. This also includes welldesigned empirical
studies and innovative automation solutions and tool architectures.
Experience papers: We invite experience papers that focus on reporting
project experience with modeldriven development. These papers should
describe the project context, detail practical lessons learned, and
provide insight about how modeldriven approaches and technologies can
be improved for application in an industrial context.
The conference tracks will include sessions on MDA, UML, MIC, and will
include papers on the following topics:
 Modeldriven development methodologies, approaches, and languages
 Model transformations
 Model Integrated Computing
 Unified Modeling Language: What are the next steps?
 Empirical studies of modeling and modeldriven development
 Tool support for any aspect of modeldriven development or model use
 Models in the development and maintenance process
 Model evaluation, formal or heuristics
 Metamodeling
 Semantics of modeling languages
 Domainspecific and concernoriented modeling
IMPORTANT DATES
Experience and Scientific Papers:
 Hard Deadline for Abstracts: March 21, 2005
 Hard Deadline for Submissions: April 4, 2005
 Notification to Authors: June 6, 2005
 Final Version of Accepted Papers: July 11, 2005
Workshop Proposals:
 Deadline for submission: May 6, 2005
 Notification of acceptance: June 24, 2005
Tutorial Proposals
 Deadline for submission: June 6, 2005
 Notification of acceptance: July 4, 2005
 Cameraready tutorial notes: September 5, 2005
Doctoral Symposium,Tools and Exhibits, Posters and Demos: TBA
PAPER SUBMISSIONS:
Submit your manuscript electronically in Postscript or PDF using the
Springer LNCS style: http://www.springer.de/comp/lncs/authors.html
Submission web site: http://sql05.sce.carleton.ca/models2005
Abstract and paper submittal will be available through the web site by
January, 2005.
Scientific papers should be no longer than 15 pages in length, and
experience papers no longer that 10 pages. Papers will undergo a
thorough process of review by a program committee comprising leading
experts from academia and industry; however, papers that are too long
may be rejected without review. Scientific proceedings will be published
by SpringerVerlag in the LNCS series, and experience papers will be
published in a companion proceedings. All papers must be original,
unpublished, and not submitted simultaneously for publication elsewhere.
Proposals for advanced workshops, tutorials and posters are requested.
See the conference web site http://www.modelsconference.org/ for details.
************************************************************************
General Chair: Stuart Kent, Microsoft, UK
Conference CoChairs: Geri Georg, Colorado State University, USA, and
Ezra Mugisa, The University of the West Indies at Mona, Jamaica
Programme Chair: Lionel Briand, Carleton University, Canada
Treasurer and Registration Chair: Robert France, Colorado State
University, USA
Experience Track Chair: Clay Williams, IBM Watson Research Center, USA
Workshop Chair: JeanMichel Bruel, University of Pau, France
Tutorials Chair: TBA
Panel Chair: Siobh=E1n Clarke, Trinity College, Ireland
Doctoral Symposium Chair: Jeff Gray, University of Alabama at
Birmingham, USA
Tool Exhibition Chair: Gunjan Mansingh, University of the West Indies at
Mona, Jamaica
Poster Chair: Felix Akinladejo, University of Technology, Jamaica
Local Arrangements Chair: Charmaine DeLisser, University of Technology,
Jamaica
Publicity Chairs: Joao Araujo, University of New Lisbon, Portugal and
Emanuel Grant, University of North Dakota, USA
Web Chair: Sudipto Ghosh, Colorado State University, USA
Program Committee: TBA
************************************************************************

*******************************************************
Lionel C. Briand, Ph.D., P. Eng.
Canada Research Chair (Tier I) in Software Quality Engineering
Carleton University
Software Quality Engineering Laboratory
Department of Systems and Computer Engineering
1125 Colonel By Drive, ME4456
Ottawa, Canada, K1S 5B6
Phone: (613) 520 2600 Ext. 2471
Fax: (613) 520 5727
Email: briand@sce.carleton.ca
URL: www.sce.carleton.ca/faculty/briand
********************************************************
Do not miss MoDELS /UML 2005: see http://www.modelsconference.org for
the 8th International Conference on Model Driven Engineering
Languages and Systems (formerly the UML series of conferences)
26Nov2004 19:47:23 0400,2758;00000000000000000000
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From: "Jacques Carette"
To:
Subject: categories: RE: Contexts for selfsimilarity
Date: Fri, 26 Nov 2004 15:00:45 0500
Organization: McMaster University
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There is one important aspect of selfsimilarity, at least as it
is understood in the context of:
> 4. Conformal/analytic: this is the natural setting for discussing
> the selfsimilarity of Julia sets of complex rational
> functions.
that seems to have been 'lost', or at least been made obscure enough =
that I
could not see it anymore.
The reason that proving selfsimilarity of some (conformal/analytic)
fractals is quite difficult is because the definitions of =
selfsimilarity
used always insist on 'bounded distortion', in other words you are =
allowed
to diform the whole before reinjecting it as a part, but the distortion =
has
to be bounded. For iterated function systems, since all the =
transformations
are linear, this is trivial to show.
But for Julia sets, since the 'natural' selfsimilarity involves =
nonlinear
transformations, proving bounded distortion is much more difficult. The
'puzzle pieces' of Yoccoz were invented explicitly to provide a tool for
showing bounded distortion. One can show that most Julia sets are
selfsimilar away for the orbits of critical points; for critical points
embedded in the Julia set, there are known dynamical conditions which =
imply
boundeddistortion, and then selfsimilarity. But there are definitely
still some open cases.
For some settings, like
> 3. Typetheoretic: recursive datatypes can be understood as
> selfsimilar objects, the bestknown example being trees
> (which can be characterized as a final coalgebra).
this bounded distortion is obvious, since there is *no* distortion at =
all.
Did I just 'miss' some condition that would insure bounded distortion? =
[I
admit to have only read Leinster's 'overview' paper, the other 2 papers =
are
on my toreadlate pile].
Jacques
26Nov2004 19:47:23 0400,1336;00000000000100000000
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Date: Fri, 26 Nov 2004 13:13:29 0600
From: Mike Oliver
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Subject: categories: Re: Open letter to F.W. Lawvere: Future of Category Theory conferences
References:
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Jiri Adamek wrote:
> We also envisage that a number of conferences with
> categorical topics would be organized independently of the activities of
> the Advisory Committee, and that the successful regional series such as
> the Peripathetic,
Hope that's "peripatetic" :)
26Nov2004 19:47:23 0400,4780;00000000000000000000
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Subject: categories: smooth spaces
To: categories@mta.ca (categories)
Date: Fri, 26 Nov 2004 12:03:46 0800 (PST)
From: "John Baez"
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Dear Categorists 
Algebraic topologists like to use not Top but a "convenient category"
of topological spaces which has the advantage of being cartesian
closed. The most popular seems to be CGHaus, the category of compactly
generated Hausdorff spaces, or "Kelly spaces". It's cartesian closed,
complete and cocomplete. I'll take these as the requirements for a
category being "convenient".
Lawvere and others have considered various topoi of "smooth spaces"
in their work on synthetic differential geometry.
I'm looking for something similar, but apparently a bit different.
I want a convenient category of smooth spaces equipped with
a forgetful functor to CGHaus:
U: Smooth > CGHaus
so I can do differential geometry and the apply it to algebraic topology
with the greatest of ease. I want U to be faithful, so that smoothness
is just a *property* of continuous maps between smooth spaces. And, I
want U to preserve limits and colimits.
The topoi used in synthetic differential geometry don't seem to
meet these requirements, because the allimportant "infinitesimal arrow"
D doesn't seem to have a good underlying Hausdorff space. You could say
its underlying Hausdorff space is the onepoint space, but this would
not give a *faithful* functor U.
Chen and Mostow have given definitions of "smooth space" that might
meet my requirements, and I'm wondering what people think of them:
K.T. Chen, Iterated paths integrals of differential forms and loop
space homology, Ann. Maths. 97 (1973) 213237.
M. A. Mostow, The differentiable space structures of Milnor classifying
spaces, simplicial complexes, and geometric realizations, J. Diff. Geom.
14 (1979), 255293.
I only have the energy to describe Chen's definition. I'm hoping
someone can give me an elegant proof that it gives a category meeting my
requirements. I think it does, but I don't know an elegant proof.
I'll use "space" to mean an object of CGHaus, though Chen actually
uses it to mean an object of Top.
Definition:
A "smooth space" X is a space equipped with, for each convex open
subset U of some R^n, a collection P(U,X) of continuous maps
f: U > X, called "plots". These need to satisfy various properties:
1) If f: U > X is a plot and g: V > U is a smooth map with V convex
open in some R^m, then fg is a plot.
2) If U is covered by convex open sets U_a and the restriction of
f: U > X to each U_a is a plot, then f is a plot.
3) If U = R^0, every map f: U > X is a plot.
Definition:
If X and Y are smooth spaces and g: X > Y is continuous, g is
defined to be "smooth" if for every plot f: U > X, gf: U > Y
is a plot.
We can try to psychoanalyze this definition:
Instead of talking about "convex open sets of R^n's" and smooth
maps between these, I'd prefer to talk about R^n's and smooth maps
between these, because I believe these form an equivalent category.
Let's call this category OpenBall.
If we do this and then drop all but condition 1), Chen's definition
would say that a smooth space is a space X equipped with a presheaf P
on OpenBall assigning to each R^n a set of "plots" P(R^n,X). He also
requires that P be a subpresheaf of the presheaf that assigns to each
R^n the set of all continuous functions from R^n to X.
Then, 2) says this presheaf P is actually a sheaf with respect to
some Grothendieck topology on OpenBall.
Then, 3) is some extra condition that guarantees smooth maps between
smooth spaces are determined by what they do to global points.
Or if you prefer, a condition that guarantees constant maps are smooth.
So, his definition seems like a way of starting with OpenBall equipped
with some Grothendieck topology, taking the category of sheaves on this,
and then forming a subcategory with the help of the functor OpenBall >
CGHaus.
I'm wondering if anyone recognizes this construction as a standard trick
for building topoi... or the composite of a couple standard tricks.
Or, if there's some similar but better way to meet my requirements!
Happy Thanksgiving,
jb
27Nov2004 13:23:18 0400,1354;00000000000000000000
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Subject: categories: Re: smooth spaces
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Somewhere I have a paper that describes classes C of spaces with the
property that Cgenerated spaces are CCC (as well as complete and
cocomplete). One of the classes is that of the finite simplexes (you
could obviously use cubes instead) which would seem to have at least the
possibility of being able to define smoothness. I am away from home now
and cannot give the exact reference, but I think it was in the Cahiers
around 1980. My recollection is that the spaces in C had to be compact,
so you could not use Euclidean spaces.
Michael
27Nov2004 13:23:18 0400,2771;00000000000000000000
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To: categories@mta.ca
Subject: categories: Re: Wilson space
InReplyTo: Message from Peter Freyd
of "Thu, 25 Nov 2004 15:44:46 EST." <200411252044.iAPKijkp004413@saul.cis.upenn.edu>
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Date: Fri, 26 Nov 2004 15:52:53 0800
From: Vaughan Pratt
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> Peter F:
> For that we need  as we called it in Cats and Alligators  Wilson space.
What section number? (Not in the index.)
> Actually,
> not the space but the linearly ordered set, most easily defined as the
> lexicographically ordered subset, W, of sequences with values in
> {1, 0, 1} consisting of all those sequences such that for all n
> a(n) = 0 => a(n+1) = 0 (take a finite word on {1,1} and pad it
> out to an infinite sequence by tacking on 0s).
(The parenthetical explanation presumably is meant to apply only to those
words containing at least one 0, i.e. W should also contain the infinite
words on {1,1} (else it would be countable).)
So shouldn't it be called the Wilson chain then, rather than Wilson space?
Ditto for the turkey chain I described yesterday (rationals in quadruplicate,
vs. triplicate for the Wilson chain).
As a topological space, I don't understand what Wilson space is. The order
interval topology (generated by the open order intervals, sets of the
form {x  p
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Date: Sat, 27 Nov 2004 20:31:59 0800 (PST)
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From: Joseph Goguen
To: categories@mta.ca
Inreplyto: <41A74055.7000205@math.ist.utl.pt> (message from Claudio Hermida
on Fri, 26 Nov 2004 14:40:21 +0000)
Subject: categories: Re: question about fibrations
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For computer scientists lurking on the list, or others who want a
really simple minded approach, the following might be helpful
@article(fun3,
title = "Some Fundamental Algebraic Tools for the Semantics of Computation,
Part 3: Indexed Categories",
author = "Andrzej Tarlecki and Rod Burstall and Joseph Goguen",
journal = "Theoretical Computer Science",
year = 1991,
volume = 91,
pages = "239264")
though of course John Gray's "Fibred and Cofibred Categories" is the urtext.
== joseph
************************************************************************
Joseph Goguen, Dept. Computer Science & Engineering, University of
California at San Diego, 9500 Gilman Drive, La Jolla CA 920930114 USA
email: jgoguen@ucsd.edu
www: http://www.cs.ucsd.edu/users/goguen/
phone: (858) 5344197 [my office]; 1246 [dept office];
7029 [dept fax]; (858) 8220702 [secy]
office: 3131 Applied Physics and Math Building
J Consciousness Studies: http://www.imprintacademic.com/jcs/
************************************************************************
> Date: Fri, 26 Nov 2004 14:40:21 +0000
> From: Claudio Hermida
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>
> > Dear Cat Community,
> >
> > I have the following question about preservation of properties of fiber
> > categories in the total category:
> >
> > Given a contravariant indexed category F : B^Op > Cat, and the
> > Grothendieck translation from the indexed category to fibrations
> > Flat(F). Then, the first projection P : Flat(F) > B forms a (split)
> > fibration, called the flattening of F.
> >
> > Now assume that the fiber categories are complete/cocomplete/adhesive.
> > Would it be the case that the total category Flat(F) of the described
> > (split) fibration is also complete/cocomplete/adhesive?
> >
> > Any references would be appreciated.
> > Thank you kindly.
> >
> > Best regards,
> > Alexey Cherchago
>
>
> Given a fibration P: Flat(F) > B, assuming B is complete then
> (E is complete and P continuous) iff P is fibred complete (complete
> fibres and continuous reindexing) .
>
> Dually for cofbirations (coming from 'covariant' indexed categories) and
> cocompleteness.
>
> I am clueless about 'adhesive categories' but Google points out to the
> following def:
> C adhesive means it has pullbacks, pushout along monos and these latter
> satisfy a certain exactness condition involving pullback stability (a
> socalled "VK square" which is actually a cube(?)).
>
> Assuming P: Flat(F) > B preserves monos, an arrow in Flat(F) is monic
> iff its image in B and its vertical factor are monic. Ergo, assume:
>
> B adhesive
> P has direct images along monos (m^* have left adjoints)
>
>
> Then, (P has fibrewise pb/po along monos) implies (Flat(F) has same and
> P preserves that).. The VK condition is a little obtruse, and at first
> sight requires the following additional assumption: the ' po along
> monos' in Flat(F) preserves cartesian morphisms and reindexing functors
> preserve po along monos. Then it seems that one gets (if one cares to do
> the relevant calculations)
>
> P fibrewise adhesive iff Flat(P) adhesive
> (for the only if, given a stack of 2 cubes where the bottom one has
> all side faces pb, the total cube is a VK square iff the top one is such).
>
> References:
> On the correspondence of limits/colimits between fibres and total
> Flat(F) see:
>
>  Gray, J. W., Fibred and Cofibred categories, Proceedings of the
> Conference on Categorical Algebra,1966.
>
> There are more recent references which treat this subject (fibred vs.
> global properties) from a more abstract point of view (and it a broader
> context), but they are probably not the first place to look if one is
> not acquainted with the basics.
28Nov2004 13:37:05 0400,1549;00000000000000000000
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Date: Sat, 27 Nov 2004 15:44:43 0500 (EST)
From: Peter Freyd
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To: categories@mta.ca
Subject: categories: Re: Wilson space
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Vaughan points out that Wilson space didn't make it into the index of
Cats & Allegators. Whoops. It's to be found in section 1.749, page 129
(but not in small caps, which, I guess, is how it was missed when
preparing the index).
It is not an orderinterval topology. Indeed, it is not Hausdorf. As I
remarked in my last post, Wilson space is definable as the final
coalgebra of the functor on Top that sends X to the scone of
X + X. (The only Hausdorf scone is the scone of the empty set.)
Its important properties are two: 1) the category of sheaves on Wilson
space is equivalent to the category of presheaves on the infinite
binary tree (viewed as a poset with the root as top); and, 2) there's
an open continuous map from the closed interval onto Wilson space
(hence from any reasonable locally euclidean space). All of which
yields a completeness theorem for intuitionistic firstorder logic
with respect to the semantics of sheaves on the reals (or any
reasonable locally euclidean space).
29Nov2004 11:21:44 0400,6038;00000000000000000000
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From: Pawel Sobocinski
Subject: categories: Adhesive categories
Date: Mon, 29 Nov 2004 12:11:04 +0100
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The subject of adhesive categories came up in this forum, first raised by
Alexey Cherchago and then googled by Claudio Hermida. I thought that I
could clear a few things up.
Adhesive categories were introduced by Steve Lack and myself in a paper
at FoSSaCS '04. Recently an extended version has been accepted for
publication in Theoretical Informatics and Applications. Both versions
are available to download from my homepage.
A van Kampen (VK) square is a certain pushout. First, such a pushout must
be stable under pullback  in the sense that if we start with a VK square
plus one morphism to the "result" of the pushout and keep taking
pullbacks, we obtain a cube where the bottom face is our original VK
square  "stable under pullback" means that the top face is now also a
pushout.
Now let us imagine that a cube is oriented in such a way that all the
arrows of the top and bottom face point out of the page. The second
property satisfied by VK squares is that starting with such an oriented
cube over a VK square where the top face is a pushout and the back faces
are pullbacks, then the front faces are required to be pullbacks.
Thus, the defining property of "VK square" can be stated reasonably
concisely (and less confusingly, if one has a picture to look at):
given any cube over a VK square with back faces pullbacks, the top face is
a pushout iff the front faces are pullbacks.
Unfortunately, not every pushout in Set is VK  which leads to the
definition of adhesive categories: a category is adhesive if it has
pushouts along monos, pullbacks and pushouts along monos are VK squares. A
weaker assumption, pushouts along regular monos being VK, defines
quasiadhesive categories. It turns out that all monos are regular in
adhesive categories. Set is adhesive, as well as any elementary topos.
Moreover, adhesivity is stable under slice, coslice, product and functor
category.
The VK property is very similar to the property satisfied by coproducts in
extensive categories. And, as in extensive categories, it can be
characterised by a suitable "equivalence of categories" definition 
perhaps a little less `abstruse' for some. Moreover, many of the cute
little lemmas about coproducts in extensive categories translate to
pushouts which led us to use the slogan "adhesive categories have
wellbehaved pushouts along monos" just as "extensive categories have
wellbehaved coproducts". Extensive and adhesive categories are also
directly related: any adhesive category with a strict initial object is
extensive. However, there are adhesive categories which are not extensive
and extensive categories which are not adhesive.
I'll also mention that the algebra of subobjects is quite nice in adhesive
categories, subobject union is calculated as the pushout along their
intersection and the resulting lattice is distributive.
Finally, perhaps I should apologise for the apparent poor quality of the
explanations in our papers. Somehow they got through 5 thorough reviews
without any of the referees being confused by the notion of VK square 
perhaps Claudio could help by suggesting how the explanation could be made
clearer.
On 26 Nov 2004, at 15:40, Claudio Hermida wrote:
>
> Given a fibration P: Flat(F) > B, assuming B is complete then
> (E is complete and P continuous) iff P is fibred complete (complete
> fibres and continuous reindexing) .
>
> Dually for cofbirations (coming from 'covariant' indexed categories)
> and
> cocompleteness.
>
> I am clueless about 'adhesive categories' but Google points out to the
> following def:
> C adhesive means it has pullbacks, pushout along monos and these latter
> satisfy a certain exactness condition involving pullback stability (a
> socalled "VK square" which is actually a cube(?)).
>
> Assuming P: Flat(F) > B preserves monos, an arrow in Flat(F) is monic
> iff its image in B and its vertical factor are monic. Ergo, assume:
>
> B adhesive
> P has direct images along monos (m^* have left adjoints)
>
>
> Then, (P has fibrewise pb/po along monos) implies (Flat(F) has same
> and
> P preserves that).. The VK condition is a little obtruse, and at first
> sight requires the following additional assumption: the ' po along
> monos' in Flat(F) preserves cartesian morphisms and reindexing functors
> preserve po along monos. Then it seems that one gets (if one cares to
> do
> the relevant calculations)
>
> P fibrewise adhesive iff Flat(P) adhesive
> (for the only if, given a stack of 2 cubes where the bottom one has
> all side faces pb, the total cube is a VK square iff the top one is
> such).
>
> References:
> On the correspondence of limits/colimits between fibres and total
> Flat(F) see:
>
>  Gray, J. W., Fibred and Cofibred categories, Proceedings of the
> Conference on Categorical
> Algebra,1966.
>
> There are more recent references which treat this subject (fibred vs.
> global properties) from a more abstract point of view (and it a broader
> context), but they are probably not the first place to look if one is
> not acquainted with the basics.
>
>
>
>
29Nov2004 16:05:09 0400,1649;00000000000100000000
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Date: Mon, 29 Nov 2004 11:00:03 0500 (EST)
From: F W Lawvere
ReplyTo: wlawvere@buffalo.edu
To: categories@mta.ca
Subject: categories: Open letter: Future of Category Theory Conferences
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Dear categorists,
In order to help insure the continuity of the international CT meetings
for the advancement of our subject, it has been proposed by Jiri Adamek,
Francis Borceux, Dominique Bourn, George Janelidze, Peter Johnstone, Bob
Rosebrugh, Jiri Rosicky, Ross Street, and Walter Tholen, that
an Advisory Committee be formed with me as chairman. I am willing to
accept this position if there are no objections.
After January 11th I will name a vice chairman and propose the
membership of the committee.
Thanking you for your confidence, I send you all best wishes for a
happy and productive new year.
Bill
************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 142602900 USA
Tel. 7166456284
HOMEPAGE: http://www.buffalo.edu/~wlawvere
************************************************************
29Nov2004 16:05:09 0400,2108;00000000000000000000
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Subject: categories: nfold categories, nfold operads
From: Stefan Forcey
To: categories@mta.ca
Date: Mon, 29 Nov 2004 12:45:01 0500 (EST)
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Dear category list,
Here is yet another operad paper that deals with issues of interest to
people in categorical homotopy. The examples mentioned in the abstract actually
make up the majority of the text, and come with illustrations.
_________________________________________________
Combinatoric nfold categories and nfold operads
Operads were originally defined as Voperads, that is, enriched in a
symmetric or braided monoidal category V. The symmetry or braiding in V is
required in order to describe the associativity axiom the operads must
obey, as well as the associativity that must be a property of the action
of an operad on any of its algebras. After a review of the role of operads
in loop space theory and higher categories we go over definitions of
iterated monoidal categories and introduce a large family of simple
examples. Then we generalize the definition of operad by defining nfold
operads and their algebras in an iterated monoidal category. We discuss
examples of these that live in the previously described categories.
Finally we describe the (n2)fold monoidal category of nfold Voperads.
_________________________________________________
It is on the arxiv at:
http://www.arxiv.org/abs/math.CT/0411561
or from my webpage:
http://www.math.vt.edu/people/sforcey/class_home/research.htm
Comments and suggestions are welcome indeed!
Thanks,
Stefan Forcey
30Nov2004 08:03:55 0400,2670;00000000000000000000
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Date: Mon, 29 Nov 2004 16:31:19 0800
From: Dusko Pavlovic
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To: categories@mta.ca
Subject: categories: Re: on Gutenberg and Mandelbrot
References: <41A40916.6070408@kestrel.edu> <9BA278923F3A11D9BEF9000393C440BC@math.tulane.edu>
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Michael Mislove wrote:
> This thread reminds me of John Rhodes' review of the classic, "The
> Algebraic Theory of Semigroups, Vol. I" by Clifford and Preston. This
> exceptional account of the state of semigroup theory in the early 60s
> contained more than copious citations, to authors for published works,
> to colleagues for ideas, and even to the extent that solutions to
> problems were cited. Rhodes, his acerbic wit at hand, commented on this
> incredible documentation of where even the simplest ideas had
> originated, was led to comment, "it's surprising that the authors
> didn't cite Gutenberg for the type."
i agree, it is easy to exaggerate with citations, and with many other
things, in all kinds of ways. one could cite gutenberg out of a
scholarly zeal, or just to emphasize that they only use the shoulders of
giants. but such generalities aside, we had a much smaller issue here:
there is this very interesting research in selfsimilarity, and it is
based on the theorem that [0,1] is a final coalgebra. on the other hand,
vaughan and i had written some papers promoting the idea that the reals
form a final coalgebra, and felt that we were a bit closer to the topic
than gutenberg to semigroups.
but enough of that.
it is interesting that mandelbrot is talking about the real interval in
coalgebraic terms. there are many such examples. continued fractions are
also a familiar instance of a coalgebraic expansion. i don't think that
we are really discovering the coalgebraic nature of selfsimilar and
analytic objects; just giving it a categorical formulation. but math
analysis consists of lots of coalgebraic constructions, often just
thinly disguised.
 dusko
30Nov2004 08:03:55 0400,2441;00000000000000000000
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Date: Tue, 30 Nov 2004 01:20:24 0000 (GMT)
Subject: categories: Re: Contexts for selfsimilarity
From: "Tom Leinster"
To:
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Dusko Pavlovic wrote:
> doesnt the 0dim vector space carry the final (terminal) coalgebra? (it
> has just one vector, so there is not much choice for v_0 and v_1.)
Nope  v_0 and v_1 have to be linearly independent. (This is like the
condition in Peter's result that the two distinguished points of the set
be distinct, without which the result degenerates; and as in Peter's
result, it can be regarded as a kind of flatness condition.)
Incidentally, I was probably wrong to suspect that the answer is something
to do with measure, as the question is posed over an arbitrary field. I
now suspect that there's a similar question whose answer has to do with
measure, but I won't attempt any further speculation here.
Tom
>>Here's a question belonging to (10), to which I don't know the
>>answer. Let C be the category whose objects are triples (V, v_0, v_1)
>> where V is a vector space and v_0 and v_1 are linearly
>>independent vectors in V, and whose maps preserve linear structure and
>> the `basepoints'. There's a `wedge' functor C x C > C
>>defined by
>>
>> (V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1)
>>
>>where (V + W)/~ is the direct sum with v_1 identified with w_0.
>>(So dim(V wedge W) = dim V + dim W  1.) There's then an
>>endofunctor G of C given by selfwedging. Question: what, if
>>any, is the terminal Gcoalgebra?
>>
>>(I suspect the answer is something to do with measure/integration 
>> again see Peter's previous postings  but really have no idea.)
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From: Dusko Pavlovic
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To: categories@mta.ca
Subject: categories: Re: Contexts for selfsimilarity
References: <1101409024.1337.65.camel@tllinux.maths.gla.ac.uk>
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doesnt the 0dim vector space carry the final (terminal) coalgebra? (it
has just one vector, so there is not much choice for v_0 and v_1.)
 dusko
Tom Leinster wrote:
>Here's a question belonging to (10), to which I don't know the
>answer. Let C be the category whose objects are triples (V, v_0,
>v_1) where V is a vector space and v_0 and v_1 are linearly
>independent vectors in V, and whose maps preserve linear structure
>and the `basepoints'. There's a `wedge' functor C x C > C
>defined by
>
> (V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1)
>
>where (V + W)/~ is the direct sum with v_1 identified with w_0.
>(So dim(V wedge W) = dim V + dim W  1.) There's then an
>endofunctor G of C given by selfwedging. Question: what, if
>any, is the terminal Gcoalgebra?
>
>(I suspect the answer is something to do with measure/integration
> again see Peter's previous postings  but really have no idea.)
>
>
>Finally, re citations: I'll stick in a PavlovicPratt reference,
>as suggested.
>
>All the best,
>Tom
>
>
>
>
>
>
>
>
>
>
>
>
>
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Date: Tue, 30 Nov 2004 03:38:00 +0100
To: categories@mta.ca
From: Agusti Roig
Subject: categories: fibrations
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******************************************
Alexey Cherchago escribi=F3:
> Dear Cat Community,
>
> I have the following question about preservation of properties of fiber
> categories in the total category:
>
> Given a contravariant indexed category F : B^Op > Cat, and the
> Grothendieck translation from the indexed category to fibrations
> Flat(F). Then, the first projection P : Flat(F) > B forms a (split)
> fibration, called the flattening of F.
>
> Now assume that the fiber categories are complete/cocomplete/adhesive.
> Would it be the case that the total category Flat(F) of the described
> (split) fibration is also complete/cocomplete/adhesive?
>
> Any references would be appreciated.
> Thank you kindly.
>
> Best regards,
> Alexey Cherchago
I'm not an expert on the field and I don't know if for split fibrations
you would have a better result, but as far as I know, if you want the
total category A of a fibered categoy \pi : A > E to be complete,
you need the fiber categories A_x to be complete, but also the base
category E .
If you want also A to be cocomplete, then you need that the fibered
category is cofibered and the base and the fibers cocomplete.
So, if you have a bifibered category (fibered and cofibered), with base
and fibers complete and cocomplete, then the total category is complete
and cocomplete (and I don't know anything about "adhesive").
Computations are easy and you can find them, for instance, in section 3 of
Model category structures in bifibred categories
JPAA 95, (1994), 203  223
But, as I said, I'm not an expert: only a user of fibered categories. So
I would also appreciate some references for Flat(F) : is this the
fibered category associated to F in the sense of SGA1, or am I missing
something?
Also another question about fibered categories: has someone developped
the notion of fibered 2category?
The reason for my question is this: I've encountered the following
situation at least four times recently. I have a 2functor
F : E > Cat
and I am interested in the (co)fibered category associated to F . So I
forget the 2structure and I take it:
\pi : A > E
Then I realize (and I need, at least in one of the examples) that A has
also a "natural" (at least in 3 of the four examples I have in mind)
structure of 2category.
So my question is: Is there any canonical way to build a fibered
2category from a 2functor?
I suspect that, in general, the answer should be "no", because in my
fourth example the 2structure of A seems very ad hoc, and has nothing
to do with F .
So maybe the right questions are:
 What should be a "fibered 2category"?
 Which extra conditions do you have to impose on F in order
to obtain a natural, canoncial, fibered 2category from it?
=20
Agust=ED Roig Mart=ED
Universitat Polit=E8cnica de Catalunya
Dept. Matem=E0tica Aplicada I, ETSEIB  FME
Diagonal 647
08028 Barcelona
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MessageID: <41AC91BF.4030109@math.ist.utl.pt>
Date: Tue, 30 Nov 2004 15:29:03 +0000
From: Claudio Hermida
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Subject: categories: Re: fibrations
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Agusti Roig wrote:
>
>
> The reason for my question is this: I've encountered the following
> situation at least four times recently. I have a 2functor
>
> F : E > Cat
>
> and I am interested in the (co)fibered category associated to F . So I
> forget the 2structure and I take it:
>
> \pi : A > E
>
> Then I realize (and I need, at least in one of the examples) that A ha=
s
> also a "natural" (at least in 3 of the four examples I have in mind)
> structure of 2category.
>
> So my question is: Is there any canonical way to build a fibered
> 2category from a 2functor?
>
> I suspect that, in general, the answer should be "no", because in my
> fourth example the 2structure of A seems very ad hoc, and has nothin=
g
> to do with F .
>
> So maybe the right questions are:
>
>  What should be a "fibered 2category"?
>  Which extra conditions do you have to impose on F in order
> to obtain a natural, canoncial, fibered 2category from it?
>
>
>
The notion of fibered 2category was introduced in (1) below=20
(2fibration). The indexed version is a "homomorphism of=20
Graycategories" F: K^coop > 2Cat and is used implicitly as such in=20
(2), which includes a brief discussion of related formulations of=20
2fibrations in the groupoidal context. Given a mere 2functor F::C >=20
Cat, one surely produces a 'covariant' version of 2fibration if one=20
does not forget anything.
define \int(F) with:
objects: (X,x) with X in C and x in FX
morphs: (f,g):(X,x) > (X',x') is f::X >X' and g:Ff(x) > x' (in FX')
2cells: a :(f,g) =3D> (f=B4,g') is a: f =3D> f' such that g' o Fa(c)=
=3D g
with the evident forgetful \pi: \int(F) > C.
References: (available from=20
http://slc.math.ist.utl.pt/claudio/publications.html)
(1) C. Hermida, {\em Some Properties of Fib as a fibred 2category\/},
in {\it Journal of Pure and Applied Algebra\/} 134 (1), 83109, 1999.
(2) C. Hermida, {\em Descent on 2fibrations and 2regular=20
2categories}, to appear in special issue of {\it Applied Categorical=20
Structures} on {\em Descent} (coproceedings of Workshop on Categorical=20
Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian=20
Categories, Fields Institute, Toronto, September 2328,=20
2002).(\textbf{in print})=20
Claudio
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Date: Tue, 30 Nov 2004 10:48:55 0800
From: Dusko Pavlovic
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oh. so then your objects are actually free algebras for the monad that
adds two new vectors to a space...
does this lead anywhere?
let the objects of *V* be sets (of base vectors). a morphism from A to B
is a linear operator from R^A to R^B (an AxB"matrix").
the monad *V*>*V* maps A > A+2, where 2 = {0,1}. it extends the
linear operators so that the adjoined constants are preserved.
let *K* be the kleisli category for ()+2.
the wedge functor V : *K* x *K* > *K* maps > A + {m} + B.
given two *K*arrows, ie linear operators Af> 0+B+1 and
Cg>0+D+1, we need to define
A+m+C fVg> 0+B+m+D+1.
conjoin the A >1 minor of f and the C>0 minor of g to get the
mcomponent ("column") of f V g.
now what might be the final coalgebra of WX = XVX?
if all of the above happens over the category *S* of sets and functions,
instead of *V* of sets and linear operators  then the final coalgebra
is (0,1). remember that this is just the generators, so when you really
make the final coalgebra by adding 2, you get [0,1]. so this is just the
freyd construction. the coalgebraic structure takes each number to one
of its binary representations.
when we work over *V*, and someone gives me a coalgebra X > WX in
*K*, ie a linear operator X > 0+X+m+X+1, this coalgebra structure
unfolds each x from X into a tree of real numbers. there are exactly
2X+3 branches coming out of each node that is not a leaf; 3 out of
those are leaves, and each of the remaining 2X nodes has 2X+3
branches coming out of it.
the final coalgebra Z > WZ would have to consist of such trees as
well, of width 2Z+3. each tree induced by x from X would have to
correspond to a unique Ztuple of real numbers.
hmm. it seems clear that Z must be wider than any X. but size is a bad
reason for something not to exist. what if we have two universes? can we
span all 2X+3trees by the vectors from some base Z?
g'night,
 dusko
Tom Leinster wrote:
>Dusko Pavlovic wrote:
>
>
>>doesnt the 0dim vector space carry the final (terminal) coalgebra? (it
>>has just one vector, so there is not much choice for v_0 and v_1.)
>>
>
>Nope  v_0 and v_1 have to be linearly independent. (This is like the
>condition in Peter's result that the two distinguished points of the set
>be distinct, without which the result degenerates; and as in Peter's
>result, it can be regarded as a kind of flatness condition.)
>
>Incidentally, I was probably wrong to suspect that the answer is something
>to do with measure, as the question is posed over an arbitrary field. I
>now suspect that there's a similar question whose answer has to do with
>measure, but I won't attempt any further speculation here.
>
>Tom
>
>
>
>>>Here's a question belonging to (10), to which I don't know the
>>>answer. Let C be the category whose objects are triples (V, v_0, v_1)
>>>where V is a vector space and v_0 and v_1 are linearly
>>>independent vectors in V, and whose maps preserve linear structure and
>>>the `basepoints'. There's a `wedge' functor C x C > C
>>>defined by
>>>
>>> (V, v_0, v_1) wedge (W, w_0, w_1) = ( (V + W)/~, v_0, w_1)
>>>
>>>where (V + W)/~ is the direct sum with v_1 identified with w_0.
>>>(So dim(V wedge W) = dim V + dim W  1.) There's then an
>>>endofunctor G of C given by selfwedging. Question: what, if
>>>any, is the terminal Gcoalgebra?
>>>
>>>(I suspect the answer is something to do with measure/integration 
>>>again see Peter's previous postings  but really have no idea.)
>>>
>
>
>
>
>
>
>
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From: Thomas Streicher
MessageId: <200411301307.iAUD7Je3030606@fb04209.mathematik.tudarmstadt.de>
Subject: categories: Re: fibrations
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Augusti Roig wrote:
> I'm not an expert on the field and I don't know if for split fibrations
> you would have a better result, but as far as I know, if you want the
> total category A of a fibered category \pi : A > E to be complete,
> you need the fiber categories A_x to be complete, but also the base
> category E .
>
> If you want also A to be cocomplete, then you need that the fibered
> category is cofibered and the base and the fibers cocomplete.
You do not need that \pi is also cofibred, it suffices that the reindexing
functors \alpha^* (for \alpha in the base) preserve limits.
Consider e.g. the category CC of all ordinals with reverse order. Then
Fam(CC) fibred over Set is a complete fibration of complete categories
but not a cofibration as otherwise each fibre would contain an initial
object (the fibre over 0 has an initial object and thus \coprod_\alpha 0
were initial in Fam(CC)(I) for \alpha : 0 > I).
Thomas