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From rrosebru@mta.ca Mon Dec 3 11:05:19 2001 -0400
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Date: Mon, 3 Dec 2001 08:49:55 +1100
To: categories@mta.ca
Subject: categories: Re: the walking adjunction and biadjunction
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James Dolan writes:
> are you sure everyone will be happy with the name "biadjunction" for the
> thing that you're talking about? i'm just vaguely wondering whether it
> might unintentionally evoke ideas about "bicategories".
Oops! I guess I fell into this trap. If biadjunction doesn't mean
bicategorical adjunction what does it mean? (I mentioned the free-living
pseudoadjunction since I thought it did.)
Steve.
From rrosebru@mta.ca Mon Dec 3 11:05:22 2001 -0400
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To: categories@mta.ca
From: Andree Ehresmann
Subject: categories: Re: the walking adjunction and biadjunction
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On the "walking adjunction"
I don't know the Pumplun's paper cited by Wyler. But there is another
reference at about the same time; indeed, the "walking adjunction" has been
explicitly constructed and studied in the paper of Auderset:
"Adjonction et monade au niveau des 2-cat=E9gories"
published in "Cahiers de Top. et Geom. Diff." XV-1 (1974), 3-20.
More formally it could also be called "the 2-sketch of an adjunction" in
the terminology in my paper with Charles Ehresmann:
"Categories of sketched structures", in the "Cahiers" XIII-2 (1972),
reprinted in
"Charles Ehresmann: Oeuvres completes et commentees" Part IV-2.
To add a remark on the terminology: When Charles introduced the concept of
a sketch (already in a Kansas report of1966, cf. "Oeuvres" Parts III-2
and IV-1), the aim was to define the 'Platonist idea' of a structure, not
only of a purely algebraic one, but also of structures like categories
(partially defined operations), fields, or even topologies. He thought
first of calling a sketch an idea, but then reserved the word "idea" for
the smallest part which helps reconstruct the sketch; for instance for a
category, the arrows which 'represent' the domain and codomain maps and the
composition law.
Sincerely
Andree C. Ehresmann
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From: "Andree Ehresmann"
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Subject: categories: Re: the walking adjunction and biadjunction
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Date: Sat, 01 Dec 2001 19:20:09 +0100
To: categories@mta.ca
From: Andree Ehresmann
Subject: categories: Re: the walking adjunction and biadjunction
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On the "walking adjunction"
I don't know the Pumplun's paper cited by Wyler. But there is another
reference at about the same time; indeed, the "walking adjunction" has been
explicitly constructed and studied in the paper of Auderset:
"Adjonction et monade au niveau des 2-cat=E9gories"
published in "Cahiers de Top. et Geom. Diff." XV-1 (1974), 3-20.
More formally it could also be called "the 2-sketch of an adjunction" in
the terminology in my paper with Charles Ehresmann:
"Categories of sketched structures", in the "Cahiers" XIII-2 (1972),
reprinted in
"Charles Ehresmann: Oeuvres completes et commentees" Part IV-2.
To add a remark on the terminology: When Charles introduced the concept of
a sketch (already in a Kansas report of1966, cf. "Oeuvres" Parts III-2
and IV-1), the aim was to define the 'Platonist idea' of a structure, not
only of a purely algebraic one, but also of structures like categories
(partially defined operations), fields, or even topologies. He thought
first of calling a sketch an idea, but then reserved the word "idea" for
the smallest part which helps reconstruct the sketch; for instance for a
category, the arrows which 'represent' the domain and codomain maps and the
composition law.
Sincerely
Andree C. Ehresmann
From rrosebru@mta.ca Mon Dec 3 20:35:59 2001 -0400
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Date: Mon, 3 Dec 2001 13:23:40 -0500 (EST)
From: F W Lawvere
Reply-To: wlawvere@acsu.buffalo.edu
To: categories@mta.ca
Subject: categories: representing adjunctions
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ONE MORE HISTORICAL CITATION
The Pumplun paper cited by Wyler as well as the Auderset paper cited
by Mme Ehresmann illustrate that the study of generic structures in
2-categories has been going on for some time. My own paper
ORDINAL SUMS AND EQUATIONAL DOCTRINES, SLNM 80 (1969) 141-155
shows that the augmented simplicial category Delta serves as the generic
monad, but moreover goes on to actually apply this to show that the
Kleisli construction is a tensor product left-adjoint to the Eilenberg-
Moore construction which is an enriched Hom. The Hom/tensor formalism
appropriate to the case of strict monoid objects is all that is required
here, as I will explain below.
AN EXTENSION AND A RESTRICTION
The important special case of FROBENIUS monads is explicitly
characterized in three ways in my paper.
Concerning the IDEMPOTENT case discussed a few days ago by Grandis
and Johnstone, note that the publication of Schanuel and Street proves
among other things that the monoid Delta in Cat has very few quotients
(see below for significance of the monoid structure).
THE GENERAL HOM/TENSOR FORMALISM AND A VERY PARTICULAR MONOID
In any cartesian-closed category with finite limits and co-limits, a
non-linear version of the Cartan-Eilenberg Hom/tensor formalism applies
to actions and biactions of monoid objects. In Cat, Delta is a (strict)
monoid and its actions are precisely monads on arbitrary categories. A
crucial part of the formalism is that categories of actions are
automatically enriched in the basic cartesian-closed category, which in
this case is Cat. There is a particular biaction of Delta, which I called
Delta plus, with the property that the enriched Hom of it into an
arbitrary Delta-action is exactly the Eilenberg-Moore category of
"algebras", automatically equipped with its structure as a Delta^op action
(co-monad). The left-adjoint tensor assigns to any category equipped with
a co-monad its Kleisli category, as a category with monad. Not only are
the calculations in this particular case quite explicit, but the enriched
Hom tensor formalism has a lot of content which is still under-exploited.
SKETCHES VERSUS PLATONISM
The often repeated slander that mathematicians think "as if" they
were "platonists" needs to be combatted rather than swallowed. What
mathematicians and other scientists use is the objectively developed human
instrument of general concepts. (The plan to misleadingly use that fact as
a support for philosophical idealism may have been an honest mistake by
Plato, or it may have been part of his job as disinformation officer for
the Athenian CIA organization; it probably would not have survived until
now had it not been for the special efforts of Cosimo de' Medici.)
It seems that a general concept has two related aspects, as I began to
realize more explicitly in connection with my paper Adjointness in
foundations, Dialectica vol. 23, 1969 281-296; I later learned that
some philosophers refer to these two aspects as
"abstract general vs. concrete general". For example, there is the
algebraic theory of rings vs. the category of all rings, or
a particular abstract group vs. the category of all permutation
representations of the group. While it is "obvious" that, at least in
mathematics, a concrete general should have the structure of a category,
because all the instances embody the same abstract general and hence
any two instances can be compared in preferred ways, by contrast it was
not until the late fifties that one realized that an abstract general can
also be construed as a category in its own right. That realization
essentially made explicit the fact that substitution is a logical
operation and indeed is the most fundamental logical operation.
Thus an abstract general is essentially a special algebraic structure
indeed a category with additional structure such as finite limits or
still richer doctrines. As with other algebraic structures there are
again two aspects, the structures themselves and their presentations which
are closely related, yet quite distinct; for example, more than one
presentation may be needed for efficient calculations determining features
of the same algebraic structure. What is meant by a presentation depends
on the doctrine: for example Delta as a mere category has an infinite
presentation used in topology, but as a strict monoidal category it has
a finite presentation.
The notion of SKETCH is the most efficient scheme yet devised for the
general construction of PRESENTATIONS OF ABSTRACT GENERALS. The fact that
particular abstract generals and the idea of sketches exist within the
historically developed objective science does not mean that they somehow
always existed; to call them "platonic" seems to detract from the honor of
their actual discoverers.
Bill Lawvere
************************************************************
F. William Lawvere
Mathematics Department, State University of New York
244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
Tel. 716-645-6284
HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere
************************************************************
From rrosebru@mta.ca Mon Dec 3 20:37:10 2001 -0400
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Date: Mon, 3 Dec 2001 12:49:52 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Re: the walking adjunction and biadjunction
Message-ID: <20011203124952.A1223@math-cl-n03.ucr.edu>
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In-Reply-To: <5.1.0.14.1.20011130181434.009ef8d0@mailx.u-picardie.fr>; from Andree.Ehresmann@u-picardie.fr on Sat, Dec 01, 2001 at 07:20:09PM +0100
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Andree Ehresmann wrote in part:
>He thought
>first of calling a sketch an idea, but then reserved the word "idea" for
>the smallest part which helps reconstruct the sketch; for instance for a
>category, the arrows which 'represent' the domain and codomain maps and the
>composition law.
There could be multiple ideas that generate the same sketch;
how do we decide which is the correct idea among equivalent ones?
OTOH, if we take equivalence classes of ideas, then we're taking sketches.
For example, one could define the idea of multiplication in a monoid
as a binary operation and a nullary operation
or alternatively as an operation on finite tuples.
The former is more common, but I prefer the latter;
who has the right idea?
-- Toby
toby@math.ucr.edu
From rrosebru@mta.ca Mon Dec 3 20:40:19 2001 -0400
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Date: Mon, 3 Dec 2001 08:50:01 -0500 (Eastern Standard Time)
From: Walter Tholen
To: categories@mta.ca
Subject: categories: Galois and Hopf 2002
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Dear Colleagues:
We are pleased to announce that a
"Workshop on Categorical Structures for Descent and Galois Theory,
Hopf Algebras and Semiabelian Categories"
will be held September 23-28, 2002, at the Fields Institute in Toronto.
For a general description of the theme of the Workshop, see below.
A second and more detailed announcement will be sent in January 2002. It
will contain a list of invited talks as well as an invitation for a
limited number of contributed talks. There is also a webpage about the
Workshop at
http://www.fields.utoronto.ca/programs/scientific/02-03/galois_and_hopf/
We hope to be able to welcome you at the Workshop.
George Janelidze (george_janelidze@hotmail.com)
Bodo Pareigis (pareigis@rz.mathematik.uni-muenchen.de)
Walter Tholen (tholen@mathstat.yorku.ca)
> Theme and Purpose of the Workshop
>
> The goal of the meeting is to spread and to advance categorical
> methods and their application amongst researchers working in three
> overlapping areas of algebra, namely in the study of
>
> (I) algebraic structures in monoidal categories and their classical
> examples, such as Hopf, Frobenius, and Azumaya algebras, and others,
> particularly those occurring in quantum field theory,
>
> (II) Galois theory vis-a-vis Grothendieck's descent theory, as well as the
> general theory of separability and decidability, applied particularly to the
> structures mentioned in (I),
>
> (III) homological algebra of non-abelian structures, such as groups, rings
> and (associative or Lie) algebras, and its extension to the structures
> mentioned in (I).
>
> The categorical methods used will include
>
> (i) 2- and higher-dimensional categorical structures, especially
> symmetric/braided monoidal categories,
>
> (ii) categorical Galois theory, monads and fibrational descent theory, and
>
> (iii) the recently developed theory of protomodular and, more specifically,
> semiabelian categories, which provides a convenient categorical setting to
> pursue classical group-theroretic and homological concepts in a very general
> context.
From rrosebru@mta.ca Mon Dec 3 20:54:58 2001 -0400
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From: Malvina Nissim
Date: Fri, 30 Nov 2001 18:39:31 GMT
Message-Id: <200111301839.SAA11756@banks.cogsci.ed.ac.uk>
To: categories@mta.ca
Subject: categories: ESSLLI 2002 Student Session
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!!! Concerns all students in Logic, Linguistics and Computer Science !!!
!!! Please circulate and post among students !!!
We apologise if you receive this message more than once.
ESSLLI-2002 STUDENT SESSION
FIRST CALL FOR PAPERS
August 5-16 2002, Trento, Italy
Deadline: February 25th, 2002
www.iccs.informatics.ed.ac.uk/~malvi/esslli02
We are pleased to announce the Student Session of the 14th European
Summer School in Logic, Language and Information (ESSLLI-2002)
organised by the Centre for scientific and technological research
(ITC-irst) in Trento and by the University of Trento, under the
auspices of the European Association for Logic, Language and
Information (FoLLI). ESSLLI-2002 will be held in Trento from August
5-16 2002. We invite submission of papers for presentation at the
ESSLLI-2002 Student Session and for appearance in the proceedings.
PURPOSE:
This seventh ESSLLI Student Session will provide, like the other
editions, an opportunity for ESSLLI participants who are students to
present their own work in progress and get feedback from senior
researchers and fellow-students. The ESSLLI Student Session
encourages submissions from students at any level, from undergraduates
(before completion of the Master Thesis) as well as postgraduates
(before completion of the PhD degree). Papers co-authored by
non-students will not be accepted. Papers may be accepted for full
presentation (30 minutes including 10 minutes of discussion) or for a
poster presentation. All the accepted papers will be published in the
ESSLLI-2002 Student Session proceedings, which will be made available
during the summer school.
KLUWER BEST PAPER AWARD: As in previous years, the best paper will be
selected by the programme committee and will be offered a prize by
Kluwer Academic Publishers to be spent on books.
REQUIREMENTS:
The Student Session papers should describe original, unpublished work,
completed or in progress that demonstrates insight, creativity, and
promise. No previously published papers should be submitted. Note
that the ESSLLI02 school will be focussed on the three main
interdisciplinary areas (Logic & Language, Logic & Computation, and
Language & Computation), while the single areas have been
dropped. Given the high interest shown over the years, the Student
Session will keep two of the single areas, namely Logic and Language,
welcoming thus submissions within the following topics: Logic,
Language, Logic & Language, Logic & Computation, Language &
Computation.
FORMAT OF SUBMISSION:
Student authors should submit an anonymous full paper headed by the
paper title, not to exceed 7 pages of length exclusive of references
and send a separate identification page (see below). Note that the
length of the final version of the accepted papers will not be allowed
to exceed 10 pages. Since reviewing will be blind, the body of the
abstract should omit author names and addresses. Furthermore,
self-references that reveal the author's identity (e.g., "We
previously showed (Smith, 1991)... ") should be avoided. It is
possible to use instead references like "Smith (1991) previously
showed...". For any submission, a plain ASCII text version of the
identification page should be sent separately, using the following
format:
Title: title of the submission
First author: firstname lastname
Address: address of the first author
......
Last author: firstname lastname
Address: address of the last author
Short summary: abstract (5 lines)
Subject area (one of): Logic | Language | Logic and
Language | Logic and Computation | Language and Computation
If necessary, the program committee may reassign papers to a more
appropriate subject area. The submission of the extended abstract
should be in one of the following formats: PostScript, PDF, RTF, or
plain text. But note that, in case of acceptance, the final version of
the paper has to be submitted in LaTeX format. Please, use A4 size
pages, 11pt or 12pt fonts, and standard margins. Submissions outside
the specified length and formatting requirements may be subject to
rejection without review.
The paper and separate identification page must be sent by
e-mail to:
malvi@cogsci.ed.ac.uk by FEBRUARY 25th 2002
ESSLLI-2002 INFORMATION: In order to present a paper at ESSLLI-2002
Student Session, at least one student author of each accepted paper
has to register as a participant at ESSLLI-2002. The authors of
accepted papers will be eligible for reduced registration fees. For
all information concerning ESSLLI-2002, please consult the ESSLLI-2002
web site at www.esslli2002.it
IMPORTANT DATES:
Deadline for submission of abstracts: February 25, 2002.
Authors Notifications: April 22, 2002.
Final version due: May 20, 2002.
ESSLLI-2002 Student Session: August 5-16, 2002.
PROGRAMME COMMITTEE:
David Ahn, University of Rochester (Language and Computation)
Carlos Areces, University of Amsterdam (Logic)
Reinhard Blutner, University of Berlin (Language)
Kees van Deemter, University of Brighton (Language and Computation)
Paul Dekker, University of Amsterdam (Logic and Language)
Juergen Dix, University of Manchester (Logic and Computation)
Marta Garcia-Matos, University of Helsinki (Logic)
Juan Heguiabehere, University of Amsterdam (Logic and Computation)
Elsi Kaiser, University of Pennsylvania (Language)
Malvina Nissim, University of Edinburgh (Chair)
Rick Nouwen, University of Utrecht (Logic and Language)
For any specific question concerning ESSLLI-2002 Student Session,
please, do not hesitate to contact me:
Malvina Nissim
ICCS, University of Edinburgh
2 Buccleuch Place, Edinburgh
EH8 9LW, UK
phone: +44 +(0)131 +650 4630
fax: +44 +(0)131 +650 6626
e-mail: malvi@cogsci.ed.ac.uk
From rrosebru@mta.ca Tue Dec 4 20:38:20 2001 -0400
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From: baez@math.ucr.edu
Message-Id: <200112040342.fB43gfM10526@math-cl-n05.ucr.edu>
Subject: categories: Sketches and Platonic Ideas
To: categories@mta.ca (categories)
Date: Mon, 3 Dec 2001 19:42:40 -0800 (PST)
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Toby Bartels writes:
> There could be multiple ideas that generate the same sketch;
> how do we decide which is the correct idea among equivalent ones?
> OTOH, if we take equivalence classes of ideas, then we're taking sketches.
> For example, one could define the idea of multiplication in a monoid
> as a binary operation and a nullary operation
> or alternatively as an operation on finite tuples.
> The former is more common, but I prefer the latter;
> who has the right idea?
I'm confused: in my understanding, a sketch basically amounts to
a way of giving generators and relations for a category with products,
Different sketches give the same category with products, not vice versa.
Your example gives two sketches, but one category with products. In
this sense, a sketch is more like an "idea" than you seem to be giving
it credit for.
By the way, in response to Lawvere's comments:
My use of the term "Platonic idea of X" for the free
category/category with products/monoidal category/2-category/whatever
on an X was not meant as an endorsement of "Platonism" in the philosophy
of mathematics - especially since "Platonism" means many things to
many people. It was also not meant to suggest that Plato had this idea.
It was basically meant to get people thinking about abstract generals
versus concrete particulars.
Best,
John Baez
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From: Michael Barr
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To: categories
Subject: categories: Re: Sketches and Platonic Ideas
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There are a number of definitions of sketch around, some of which require
it to be a category with finite products. In one of Ehresmann's (and
Bastiani's, I believe) there is mentioned the possibility of its being
what they called a quasicategory (or some such substructure term) in which
composition is a partly defined multi-ary operation (in other words, fgh
could be defined without fg or gh being defined). Charles and I realized
that this was equivalent to what we called a graph with diagrams, which
seemed a more useable notion. So what we called a sketch was a graph with
diagrams as well as certain cones and cocones that were singled out to be
taken to limits and colimits, resp. Peter Johnstone criticized us for
doing the equivalent of replacing groups by generators and relations,
which is correct, but it was a conscious decision and there were reasons
for it. I had never heard the term "idea" in this connection or we might
have used it. But anyway, "sketch" is used in different ways and I guess
Charles and I contributed to this, but didn't create it.
On Mon, 3 Dec 2001 baez@math.ucr.edu wrote:
> Toby Bartels writes:
>
> > There could be multiple ideas that generate the same sketch;
> > how do we decide which is the correct idea among equivalent ones?
> > OTOH, if we take equivalence classes of ideas, then we're taking sketches.
> > For example, one could define the idea of multiplication in a monoid
> > as a binary operation and a nullary operation
> > or alternatively as an operation on finite tuples.
> > The former is more common, but I prefer the latter;
> > who has the right idea?
>
> I'm confused: in my understanding, a sketch basically amounts to
> a way of giving generators and relations for a category with products,
> Different sketches give the same category with products, not vice versa.
> Your example gives two sketches, but one category with products. In
> this sense, a sketch is more like an "idea" than you seem to be giving
> it credit for.
>
> By the way, in response to Lawvere's comments:
>
> My use of the term "Platonic idea of X" for the free
> category/category with products/monoidal category/2-category/whatever
> on an X was not meant as an endorsement of "Platonism" in the philosophy
> of mathematics - especially since "Platonism" means many things to
> many people. It was also not meant to suggest that Plato had this idea.
> It was basically meant to get people thinking about abstract generals
> versus concrete particulars.
>
> Best,
> John Baez
>
>
>
>
>
>
From rrosebru@mta.ca Wed Dec 5 15:25:53 2001 -0400
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Subject: categories: Re: the walking adjunction and biadjunction
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Toby Bartels wrote:
>
> For example, one could define the idea of multiplication in a monoid
> as a binary operation and a nullary operation
> or alternatively as an operation on finite tuples.
> The former is more common, but I prefer the latter;
> who has the right idea?
An interesting question in itself. I don't think either idea is "right", but
I (presumably) share with you the feeling that often the latter is more
appropriate. However, if you resolve wholeheartedly never to use a binary +
nullary presentation of a monoid-like structure then you actually find
yourself in quite an extreme position. For instance, a monoid would be
defined as a set M together with an n-fold operation
(m_1, ..., m_n) |---> [m_1 ... m_n]
on M for each natural n, subject to axioms. This is as expected so far, but
we've disallowed ourselves from using what would probably be the natural
choice of axioms,
[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
m = [m],
since this is a binary + nullary presentation. So instead we should derive
from the n-fold multiplications a k-ary operation o_T on M for each (finite,
planar) k-leafed tree T; and the axioms then become that o_T = o_U for any
two k-leafed trees T and U.
The situation gets more extreme still if you want a wholeheartedly
non-binary-and-nullary presentation of the notion of monoidal category. We
have an underlying category M, an n-fold tensor functor for each n, and then
coherence cells obeying coherence axioms. The obvious choice for the
coherence cells comes from turning the two equations above into specified
isomorphisms, but again this is disallowed, so we have to specify a coherence
cell o_T --~--> o_U for each T and U, where o_T, o_U are now derived tensor
functors. Then we need to put axioms on the coherence cells, and once more
the obvious way of doing this involves something of a binary + nullary
character. Specifically, you have to make sure that the coherence cells o_T
--~--> o_U are compatible with "grafting of trees", which means taking a
k-leafed tree T and sticking onto its leaves k trees T_1, ..., T_k, to make a
new tree T(T_1, ..., T_k) - but this expression has *2* (bad number!) levels
of trees. So we need to replace these axioms with equivalent
non-binary-and-nullary ones, and this means considering more complicated
structures still.
(The considerations in the last paragraph are really to do with writing down
a non-binary-and-nullary presentation of the theory of operads, which are
themselves monoids of a sort.)
Tom
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From: "F. William Lawvere"
To: categories@mta.ca
Subject: categories: Re: Sketches and Platonic Ideas
Date: Wed, 05 Dec 2001 04:36:21
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Certainly I did not mean to suggest that either John or Andree were
supporting platonism as a philosophy of mathematics. In fact I had
momentarily even forgotten that John had used the term. In my 1972
Perugia Notes I had made an attempt to characterize the relation between
these sorts of mathematical considerations and philosophy by saying that
while platonism is wrong on the relation between Thinking and Being,
something analogous is correct WITHIN the realm of Thinking. The relevant
dialectic there is between abstract general and concrete
general.
Not concrete particular ("concrete" here does not mean
"real").There is another crucial dialectic making particulars
(neither abstract nor concrete) give rise to an abstract
general; since experiments do not mechanically give rise to theory, it is
harder to give a purely mathematical outline of how that dialectic
works, though it certainly does work. A mathematical model of it can be
based on the hypothesis that a given set of particulars is somehow itself
a category (or graph), i.e., that the appropriate ways of comparing the
particulars are given but that their essence is not. Then their
"natural structure" (analogous to cohomology operations) is an
abstract general and the corresponding concrete general receives a
Fourier-Gelfand-Dirac functor from the original particulars. That
functor is usually not full because the real particulars are infinitely
deep and the natural structure is computed with respect to some
limited doctrine; the doctrine can be varied, or "screwed up or down" as
James Clerk Maxwell put it, in order to see various
phenomena.
From: baez@math.ucr.edu
>To: categories@mta.ca (categories)
>Subject: categories: Sketches and Platonic Ideas
>Date: Mon, 3 Dec 2001 19:42:40 -0800 (PST)
>
>Toby Bartels writes:
>
>> There could be multiple ideas that generate the same sketch;
>> how do we decide which is the correct idea among equivalent ones?
>> OTOH, if we take equivalence classes of ideas, then we're taking sketches.
...
>> who has the right idea?
>
> I'm confused: in my understanding, a sketch basically amounts to
...
>By the way, in response to Lawvere's comments:
>
> My use of the term "Platonic idea of X" for the free
...
>versus concrete particulars.
>Best,
>John Baez
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Date: Wed, 05 Dec 2001 15:52:29 -0500
To: categories@mta.ca
From: Charles Wells
Subject: categories: Sketches
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This is in reply to Toby Bartels, quoted below. I don't believe that those
of us who have written about "ideas" in Ehresmann's sense ever conceived
that each theory (sketch) was based on one right idea. There is no
"correct" idea for a given sketch.
I want to add, for those new to the subject, that the word "sketch" has
been used with at least three meanings. Ehresmann and his students use it
for a structure which is a weakening of the concept of category (the
composite may not be defined for all composable pairs) plus specified cones
and/or cocones. Many others have used the word sketch to refer to a
category with specified cones and/or cocones. Michael Barr and I in our
two books used "sketch" to mean a graph with specified cones and/or cocones
plus some commutativity conditions on paths; that is in the same spirit as
Ehresmann's "idea".
--Charles Wells
>Andree Ehresmann wrote in part:
>
> >He thought
> >first of calling a sketch an idea, but then reserved the word "idea" for
> >the smallest part which helps reconstruct the sketch; for instance for a
> >category, the arrows which 'represent' the domain and codomain maps and the
> >composition law.
>
>There could be multiple ideas that generate the same sketch;
>how do we decide which is the correct idea among equivalent ones?
>OTOH, if we take equivalence classes of ideas, then we're taking sketches.
>For example, one could define the idea of multiplication in a monoid
>as a binary operation and a nullary operation
>or alternatively as an operation on finite tuples.
>The former is more common, but I prefer the latter;
>who has the right idea?
>
>
>-- Toby
> toby@math.ucr.edu
Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
Send all mail to:
105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles@freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
genealogical website:
http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
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Date: Thu, 6 Dec 2001 00:23:14 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Sketches and ideas (Was: the walking adjunction and biadjunction)
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Tom Leinster wrote in part:
>Toby Bartels wrote:
>>For example, one could define the idea of multiplication in a monoid
>>as a binary operation and a nullary operation
>>or alternatively as an operation on finite tuples.
>>The former is more common, but I prefer the latter;
>>who has the right idea?
>An interesting question in itself. I don't think either idea is "right",
That was supposed to be my point.
Just as a group can be described many ways by generators and relations,
so a sketch (if we define a sketch to be an entire category;
apparently that varies) can be described many ways by ideas.
It's the category that truly characterises what a monoid is
(in the given doctrine), so it better deserves the name "idea",
if we're trying to hark back to Plato-n (even just to be cute).
(Whether or not it's too late to change, I can't say.)
>we've disallowed ourselves from using what would probably be the natural
>choice of axioms,
>[[m_1^1 ... m_1^{k_1}] ... [m_n^1 ... m_n^{k_n}]] = [m_1^1 ... m_n^{k_n}],
>m = [m],
>since this is a binary + nullary presentation.
2 indices and 0 indices. *Gulp* You're right!
I always felt annoyed having to write in that nullary axiom; now I know why.
>So instead we should derive
>from the n-fold multiplications a k-ary operation o_T on M for each (finite,
>planar) k-leafed tree T; and the axioms then become that o_T = o_U for any
>two k-leafed trees T and U.
I suppose that you're aware of this, but note that we need to allow
nodes that don't branch but also aren't labelled (considered leaves),
which is where we place []. For example, the tree
m .
\ /
\ /
\./
indicates the product [m[]]; that it equals m is the right unit law.
This threw me for a moment, since [] seemed at first to have disappeared.
We could also go straight to trees and define them as the basic operations,
then requiring as axiom that grafting of trees produces the same result
as composing the operations.
If we were defining a nonassociative operation without identity,
then we could denote the basic operations by *binary* trees.
>Specifically, you have to make sure that the coherence cells o_T
>--~--> o_U are compatible with "grafting of trees", which means taking a
>k-leafed tree T and sticking onto its leaves k trees T_1, ..., T_k, to make a
>new tree T(T_1, ..., T_k) - but this expression has *2* (bad number!) levels
>of trees.
The nullary counterpart is grafting 0 trees to get the tree m.
>So we need to replace these axioms with equivalent
>non-binary-and-nullary ones, and this means considering more complicated
>structures still.
Well, I managed to introduce grafting back before the categorification!
Aren't I clever? Too clever for my own good? ^_^
>(The considerations in the last paragraph are really to do with writing down
>a non-binary-and-nullary presentation of the theory of operads, which are
>themselves monoids of a sort.)
As long as we learn the lesson that binary operations
warrant a search for their nullary partners,
then we've done the important thing, at least,
even if we miss out on some ever more complicated elegance.
-- Toby
toby@math.ucr.edu
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for categories-list@mta.ca; Thu, 06 Dec 2001 16:58:49 -0400
Date: Thu, 6 Dec 2001 09:00:12 -0500 (EST)
From: Oswald Wyler
To:
Subject: categories: Reference wanted
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I'm sure that the following is known, but I've never seen it in print.
Does someone have a reference for it?
Proposition. Let U be a monadic functor, in the sense of Mac Lane's CWM.
If U factors U=HG with H faithful and amnestic, and G having a left adjoint,
then G is monadic.
From rrosebru@mta.ca Thu Dec 6 17:03:41 2001 -0400
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Date: Thu, 06 Dec 2001 08:28:52
To: categories@mta.ca
From: Colin McLarty
Subject: categories: Two Days
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Working on the history of category theory I find that Mahlon Marsh Day
hired several category theorists at the University of Illinois
Champaign-Urbana in the 1960s. I would like to know whatever people can
tell me about his connections to category theory--perhaps through Eilenberg?
Also, does anyone here know whether Mahlon Michael Day was Mahlon Marsh
Day's son? Mahlon Michael Day got a PhD at Chicago in 1967 with Kaplansky.
Thanks, Colin
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Date: Thu, 06 Dec 2001 15:58:53 -0500
To: categories@mta.ca
From: Charles Wells
Subject: categories: Defining monoids
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In talking about defining monoids, Toby Bartels wrote:
"We could also go straight to trees and define them as the basic operations,
then requiring as axiom that grafting of trees produces the same result
as composing the operations."
This is the mu operation of the corresponding monad. Every single-sorted
"idea" in the sense of the recent discussion generates a monad in sets with
a mu like this. For each set S there is a set of possible computations TS,
a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica
that says the computation consisting of a single node results in that node;
these subject to the monad laws. In other words, the phenomenon you noted
is an instance of a general result.
--Charles Wells
Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
Send all mail to:
105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles@freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
genealogical website:
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From: Martin Escardo
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To: categories@mta.ca
Subject: categories: CFP: Workshop Domains VI
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Call for abstracts
Domains VI
Birmingham, 16-19 September 2002.
The Workshop on Domains is aimed at computer scientists and
mathematicians alike who share an interest in the mathematical
foundations of computation. The workshop will focus on domains, their
applications and related topics. Previous meetings were held in
Darmstadt (94,99), Braunschweig (96), Munich (97) and Siegen (98). The
emphasis is on the exchange of ideas between participants similar in
style to Dagstuhl seminars.
INVITED SPEAKERS
Ulrich Berger University of Wales Swansea
Thierry Coquand Goeteborg University
Jimmie Lawson Louisiana State University
John Longley University of Edinburgh
* Dag Normann University of Oslo
Prakash Panangaden McGill University
Uday Reddy University of Birmingham
Thomas Streicher Darmstadt University
* to be confirmed
SCOPE
Domain theory has had applications to programming language
semantics and logics (lambda-calculus, PCF, LCF), recursion theory
(Kleene-Kreisel countable functionals), general topology (injective
spaces, function spaces, locally compact spaces, Stone duality),
topological algebra (compact Hausdorff semilattices) and analysis
(measure, integration, dynamical systems). Moreover, these
applications are related - for example, Stone duality gives rise to a
logic of observable properties of computational processes.
As such, domain theory is highly interdisciplinary. Topics of
interaction with domain theory for this workshop include, but are not
limited to:
program semantics
program logics
probabilistic computation
exact computation over the real numbers
lambda calculus
games
models of sequential computation
constructive mathematics
recursion theory
realizability
real analysis
topology
locale theory
metric spaces
category theory
topos theory
type theory
SUBMISSION OF ABSTRACTS
One-page abstracts should be submitted to
domainsvi@cs.bham.ac.uk
Shortly after an abstract is submitted (usually one or two weeks),
the authors will be notified by the programme committee. Abstracts
will be dealt with on a first-come/first-served basis.
DEADLINE
30 April 2002
ACCOMODATION
We meeting will take place at "The Manor House" halls of residence
of the University of Birmingham (http://www.bham.ac.uk/conferences/#Manor).
More details will be provided at a later date.
PROGRAMME COMMITTEE
Martin Escardo University of Birmingham
Achim Jung University of Birmingham
Klaus Keimel Darmstadt University
Alex Simpson University of Edinburgh
ORGANIZATION COMMITTEE
Martin Escardo University of Birmingham
Achim Jung University of Birmingham
PUBLICATION
We plan to publish proceedings of the workshop in lecture notes
series. There will be a call for papers after the workshop takes
place. The papers will be refereed according to normal publication
standards.
URL
http://www.cs.bham.ac.uk/~wd6/index.html
From rrosebru@mta.ca Fri Dec 7 20:08:48 2001 -0400
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To: categories@mta.ca
From: grandis@dima.unige.it (Marco Grandis)
Subject: categories: Preprint: Directed homotopy theory, II. Homotopy constructs,
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The following preprint is available:
M. Grandis,
Directed homotopy theory, II. Homotopy constructs,
Dip. Mat. Univ. Genova, Preprint 446 (Dec 2001).
(19 p.)
___
Abstract.
Directed Algebraic Topology studies phenomena where privileged directions
appear, derived from the analysis of concurrency, traffic networks,
space-time models, etc.
This is the sequel of a paper, 'Directed homotopy theory, I. The
fundamental category', where we introduced directed spaces, their non
reversible homotopies and their fundamental category. Here we study some
basic constructs of homotopy, like homotopy pushouts and pullbacks, mapping
cones and homotopy fibres, suspensions and loops, cofibre and fibre
sequences.
___
Part I and II are available, in ps:
ftp://www.dima.unige.it/Home/grandis/public/Dht1.ps
ftp://www.dima.unige.it/Home/grandis/public/Dht2.ps
___
Marco Grandis
Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: grandis@dima.unige.it
tel: +39.010.353 6805 fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/
ftp://www.dima.unige.it/Home/grandis/public/
From rrosebru@mta.ca Fri Dec 7 20:08:48 2001 -0400
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Date: Fri, 7 Dec 2001 12:30:42 +0100
To: categories@mta.ca
From: Giovanni Sambin
Subject: categories: 2WFTop: information update
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information update about the:
SECOND WORKSHOP ON FORMAL TOPOLOGY
Auditorium S. Margherita, Campo S. Margherita
Venice, April 4-6, 2002
The first announcement can now be found, together with updated information, on the web page http://www.math.unipd.it/~logic/wftop2
Here follow the main variations with respect to the first announcement:
Invited speakers. The list of invited speakers now includes Bernhard Banaschewski, Martin Escardo, Henri Lombardi, Christopher Mulvey, Peter Johnstone, Erik Palmgren, Mike Smyth, Steve Vickers.
Tutorials. One day, 3 April 2002, of tutorials by Peter Aczel, Bernhard Banaschewski, Giovanni Sambin and Silvio Valentini.
Accomodation. Due to the number of requests, we booked two buildings, so that now the safe deadline for booking convenient accomodation is extended to 31 December 2001.
Important dates:
December 31, 2001 : early registration (with safe accomodation)
February 28, 2002: deadline for the submission of papers
March 15, 2002: program is decided
April 3, 2002: tutorials
April 4 - 6, 2002: workshop
April 7: trip by private boat on the lagoon
Registration and accomodation. The form below must be sent to Damiano Macedonio, mace@dsi.unive.it.
Name and Family name:
Institution:
Address:
E-mail address:
Date of arrival: Date of departure:
Kind of accomodation (if required):
low cost (Palazzo Zenobio, room with 3-4 beds, 15-30 euros each person)
double rooms (Fondazione Levi or Palazzo Zenobio, with bathroom, 40-60 euros each person)
single rooms (a very limited number of them is available)
From rrosebru@mta.ca Fri Dec 7 20:08:51 2001 -0400
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Date: Fri, 07 Dec 2001 11:09:23
To: categories@mta.ca
From: Colin McLarty
Subject: categories: Why exact categories? history
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Myles Tierney has told me about a perspective on exact categories,
in the 1960s, that I had not understood. I probably still do not see it
very much the way it looked then. So I ask about it here. What were the
reasons for studying exact categories in the 1960s?
Here is what I used to think: Every additive category with a
generator has a faithful functor to the category of Abelian groups.
MacLane had explored this idea in 1950. Then Grothendieck's Tohoku paper
axiomatized Abelian categories in a more useful way for homological
algebra, and showed that all sheaf categories satisfied the axioms (i.e.
sheaves of Abelian groups on topological spaces, and the key theorem says
they have enough injectives). That created two reasons to look for a
non-additive generalization. First, to extend from Abelian groups to all
groups, for use in non-Abelian cohomology. (MacLane had already hinted at
replacing Abelian groups by all groups in 1950). And second to axiomatize
sheaves of sets. The exact category axioms were a promising non-additive
analogue to the Abelian category axioms.
And I have always thought of the Abelian category embedding
theorems as proving that, if you want to, you can think of Abelian
categories as concrete categories with the natural limits and colimits.
Myles did not disagree with any of that but he put it this way:
Not all categories enriched in Abelian groups are so nicely embeddable in
the category of Abelian groups, but the Abelian categories are.
This suggested a general question, when does an enriched category
embed nicely in the enriching category? And Myles had a good description
of which Abelian-group enriched categories are Abelian: the exact ones. So
the exact category axioms became an approach to this problem.
To me this question seems very different from looking for a
non-additive analogue of Abelian categories. Am I wrong about that? How
did this question look in, say, 1970? How did it look at Dalhousie?
Thanks, Colin
From rrosebru@mta.ca Fri Dec 7 20:08:52 2001 -0400
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From: S.J.Vickers@open.ac.uk
To: categories@mta.ca
Subject: categories: Two constructivity questions
Date: Fri, 7 Dec 2001 10:55:29 -0000
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Does any one know the answers to these questions?
1. Is trigonometry valid in toposes? (I'll be astonished if it isn't.)
2. Does a polynomial over the complex field C have only finitely many roots?
More precisely:
1. Over any topos with nno, let R be the locale of "formal reals", i.e. the
classifier for the geometric theory of Dedekind sections.
Do sin, cos, arctan, etc. : R -> R exist and satisfy the expected
properties? Are there general results (e.g. on power series) that say Yes,
of course they do?
2. Consider the space S of square roots of the generic complex number.
Working over C, it is the locale corresponding to the squaring map s: C ->
C, z |-> z^2. The fibre over w is the space of square roots of w.
s is not a local homeomorphism, so S is not a discrete locale. Hence we
can't say S is even a set, let alone a finite set in any of the known
senses. I don't believe its discretization pt(S) is Kuratowski finite
either. If I've calculated it correctly, it is S except for having an empty
stalk over zero (oops!), and there is no neighbourhood of zero on which an
enumeration can be given of all the elements of pt(S).
On the other hand, S is a Stone locale - one can easily construct the sheaf
of Boolean algebras that is its lattice of compact opens. That sheaf of
Boolean algebras is not Kuratowski finite, nor even, it seems to me, a
subsheaf of a Kuratowski finite sheaf.
So is there any sense at all in which S is finite?
Steve Vickers
Department of Pure Maths
Faculty of Maths and Computing
The Open University
-----------
Tel: 01908-653144
Fax: 01908-652140
Web: http://mcs.open.ac.uk/sjv22
From rrosebru@mta.ca Fri Dec 7 20:08:54 2001 -0400
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From: Michael Barr
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Subject: categories: Re: Two Days
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I knew Mahlon Day at Urbana quite well. If he had a mathematician son, I
was unaware of it, but you should ask John Gray. Now as far as I know he
didn't hire so many category theorists. For example, I was hired by Paul
Bateman. So was Jon Beck. As far as I can tell, the only one hired by
Mahlon was John Gray. And he may have known Eilenberg, but only as one
mathematician may know another. But your entire question is based on a
misconception that the hiring at Illinois was based on any sort of plan.
The fact is that there was a severe shortage of mathematicians in those
days and UI was hiring a couple dozen people a year (and losing a similar
number) and anyone who was publishing or well recommended and showed any
interest was getting offers. I got at least one utterly unsolicited firm
offer from a school I had had no contact with. Possibly, probably,
someone like Sammy had given them my name when asked, but that is all.
And I received a number of invitations to apply for a job and did and got
an offer (and a pay raise from UI). So they didn't even ask what kind of
math you did, only that you did some kind of math. Hard to believe what
it was like in those days.
The people at Illinois who were there for the first midwest category
meeting in 1965 were Alex Heller (an algebraic topologist, with an
interest in category theory), John Gray (category theorist), Jon Beck
(algebraic topology & category theory), Max Kelly (there on a one year
leave, category theory), and me (homological algebra). In fact, I wasn't
even invited originally; I can thank Max for telling Saunders to invite
me. My interest in category theory developed later at the ETH. Now
Eckmann is probably the one person most responsible for bringing category
theorists together.
A history of category theory should talk not only about the founding
fathers, but also a certain number of godfathers, people who were not
category theorists themselves, but strongly encouraged it. I would
include (but not limit it to) Beno Eckmann, Peter Hilton, Alex Heller,
David Harrison, .... Of course, there were a number of others who while
not primarily category theorists made actual contributions to category
theory: Grothendieck, Dan Kan, Albrecht Dold and Dieter Puppe,...
Above all, Colin, one should not write a history of category theory
without interviewing as many of these people as are still alive and it is
damned shame that no one has done this till now. For Eilenberg and
Harrison, it is already too late.
Michael
On Thu, 6 Dec 2001, Colin McLarty wrote:
> Working on the history of category theory I find that Mahlon Marsh Day
> hired several category theorists at the University of Illinois
> Champaign-Urbana in the 1960s. I would like to know whatever people can
> tell me about his connections to category theory--perhaps through Eilenberg?
>
> Also, does anyone here know whether Mahlon Michael Day was Mahlon Marsh
> Day's son? Mahlon Michael Day got a PhD at Chicago in 1967 with Kaplansky.
>
> Thanks, Colin
>
>
>
>
From rrosebru@mta.ca Sat Dec 8 09:51:26 2001 -0400
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Date: Sat, 8 Dec 2001 10:22:52 +0000 (GMT)
From: "Dr. P.T. Johnstone"
To: categories@mta.ca
Subject: categories: Re: Two constructivity questions
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On Fri, 7 Dec 2001 S.J.Vickers@open.ac.uk wrote:
> Does any one know the answers to these questions?
>
> 1. Is trigonometry valid in toposes? (I'll be astonished if it isn't.)
> 2. Does a polynomial over the complex field C have only finitely many roots?
>
> More precisely:
>
> 1. Over any topos with nno, let R be the locale of "formal reals", i.e. the
> classifier for the geometric theory of Dedekind sections.
>
> Do sin, cos, arctan, etc. : R -> R exist and satisfy the expected
> properties? Are there general results (e.g. on power series) that say Yes,
> of course they do?
The space of Dedekind reals is Cauchy-complete, so any convergent
power series such as sin or cos defines an endomorphism of it.
Moreover, provided (as in this case) we can calculate a "modulus of
convergense" for the power series explicitly from a bound for x, it's easy
to see that the construction x |--> sin x commutes with inverse image
functors, so it must be induced by an endomorphism of the classifying
topos (that is, of the locale of formal reals).
>
> 2. Consider the space S of square roots of the generic complex number.
> Working over C, it is the locale corresponding to the squaring map s: C ->
> C, z |-> z^2. The fibre over w is the space of square roots of w.
>
> s is not a local homeomorphism, so S is not a discrete locale. Hence we
> can't say S is even a set, let alone a finite set in any of the known
> senses. I don't believe its discretization pt(S) is Kuratowski finite
> either. If I've calculated it correctly, it is S except for having an empty
> stalk over zero (oops!), and there is no neighbourhood of zero on which an
> enumeration can be given of all the elements of pt(S).
>
> On the other hand, S is a Stone locale - one can easily construct the sheaf
> of Boolean algebras that is its lattice of compact opens. That sheaf of
> Boolean algebras is not Kuratowski finite, nor even, it seems to me, a
> subsheaf of a Kuratowski finite sheaf.
>
> So is there any sense at all in which S is finite?
>
That's a good question. I've never thought about notionss of finiteness
for non-discrete locales (someone should!). For the set of points of S,
I believe it should be what Peter Freyd called "R-finite" ("R" for
"Russell"): intuitively, this means that there is a bound on the size
of its K-finite subsets. (However, I don't have a proof of this.)
R-finiteness is quite a lot weaker than \tilde{K}-finiteness (being
locally a subobject of a K-finite object), but it's still a reasonably
well-behaved notion of finiteness (e.g. it is preserved by functors
which preserve all finite limits and colimits).
Peter Johnstone
From rrosebru@mta.ca Sun Dec 9 19:23:42 2001 -0400
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Date: Sun, 9 Dec 2001 15:20:47 +0000 (GMT)
From: "Dr. P.T. Johnstone"
To: Categories mailing list
Subject: categories: Constructive finiteness
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It is not constructively true, as I conjectured yesterday, that
the set of roots of a polynomial over C is Russell-finite.
Let X be the subspace of C consisting of 0 and all points whose
argument is a rational multiple of \pi, and consider the sheaf of
solutions of z^2 - f = 0, where f: X --> C is the inclusion map.
It is easy to see that the stalk of this sheaf at 0 is
uncountably infinite. Since R-finiteness is preserved by
inverse image functors, this yields a counterexample.
This doesn't, of course, answer Steve Vickers' original question
whether there is a sense in which the *locale* of roots of a
polynomial can be said to be finite. But it does indicate that
the appropriate notion of finiteness, if it exists, must be
a rather delicate one.
Peter Johnstone
From rrosebru@mta.ca Sun Dec 9 19:23:45 2001 -0400
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Date: Sun, 09 Dec 2001 10:35:26 +0000
To: categories@mta.ca
From: S Vickers
Subject: categories: Re: Two constructivity questions
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As before, let S be the Stone locale of square roots of the generic complex
number. The question is, In what sense can S be considered finite?
Here is one idea that occurs to me.
If a set is acted on transitively by a finite group, then classically it
must be finite (and I dare say some constructive statement of this is also
true).
S is acted on by the discrete group {+1, -1} (by multiplication in C).
Hence if that action can be considered transitive in some way, that would
be a finiteness property of S (or, rather, finiteness _structure_ on S).
If a: S x {+1, -1} -> S is the action, then I believe I can prove (by
techniques involving the upper powerlocale) that
: S x {+1, -1} -> S x S
is a proper surjection. This would seem to be a natural way to capture
transitivity of a and hence a finiteness property of S.
More generally, if an action on a locale by a finite group has only
finitely many orbits (using the above idea to specify transitivity on the
orbits), then that would be a finiteness property of the locale.
One might ask whether, by Galois theory, this can be applied to arbitrary
polynomials over C.
Steve Vickers.
From rrosebru@mta.ca Mon Dec 10 10:17:29 2001 -0400
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Date: Sun, 9 Dec 2001 20:35:20 -0500 (EST)
From: JAMES STASHEFF
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To: categories@mta.ca
Subject: categories: Re: Defining monoids
In-Reply-To: <5.1.0.14.2.20011206155739.0204cd50@mail.oberlin.net>
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For those who prefer to see the forests and the trees,
that point of view is prominent in the operad/modad/monoidal interaction.
Much such material will be in our book
on Operads (Markl and Shnider and me)
.oooO Jim Stasheff jds@math.unc.edu
(UNC) Math-UNC (919)-962-9607
\ ( Chapel Hill NC FAX:(919)-962-2568
\*) 27599-3250
http://www.math.unc.edu/Faculty/jds
On Thu, 6 Dec 2001, Charles Wells wrote:
>
>
> In talking about defining monoids, Toby Bartels wrote:
>
> "We could also go straight to trees and define them as the basic operations,
> then requiring as axiom that grafting of trees produces the same result
> as composing the operations."
>
> This is the mu operation of the corresponding monad. Every single-sorted
> "idea" in the sense of the recent discussion generates a monad in sets with
> a mu like this. For each set S there is a set of possible computations TS,
> a mu:TTS to TS, and a "OneIdentity" operation in the sense of Mathematica
> that says the computation consisting of a single node results in that node;
> these subject to the monad laws. In other words, the phenomenon you noted
> is an instance of a general result.
>
> --Charles Wells
>
> Charles Wells,
> Emeritus Professor of Mathematics, Case Western Reserve University
> Affiliate Scholar, Oberlin College
> Send all mail to:
> 105 South Cedar St., Oberlin, Ohio 44074, USA.
> email: charles@freude.com.
> home phone: 440 774 1926.
> professional website: http://www.cwru.edu/artsci/math/wells/home.html
> personal website: http://www.oberlin.net/~cwells/index.html
> genealogical website:
> http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
> NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
>
>
>
>
>
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From: Chin Wei Ngan
Message-Id: <200112121138.TAA20037@sunA.comp.nus.edu.sg>
Subject: categories: CFP : ASIA-PEPM 2002
To: categories@mta.ca
Date: Wed, 12 Dec 2001 19:38:53 +0800 (GMT-8)
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CALL FOR PAPERS
ACM SIGPLAN ASIAN Symposium on
Partial Evaluation and Semantics-Based Program Manipulation (ASIA-PEPM'02)
Aizu, JAPAN, September 12-14 2002
(co-located with FLOPS2002)
http://www.comp.nus.edu.sg/asia-pepm02
Submission deadline: 1st March 2002
The ASIA-PEPM'02 symposium will bring together researchers working in the
areas of semantics-based program manipulation, partial evaluation, and
program analysis. The symposium focuses on techniques, supporting
theory, and applications for the analysis and manipulation of programs.
Technical topics include, but are not limited to:
* Program manipulation techniques: transformation, specialization,
normalization, reflection, rewriting, run-time code generation,
multi-level programming.
* Program analysis techniques: abstract interpretation, static
analysis, binding-time analysis, type-based analysis.
* Related issues in language design and models of computation:
imperative, functional, logical, constraint-based, object-oriented,
parallel, concurrent, secure, domain-specific.
* Programs as data objects: staging, meta-programming, incremental
computation, mobility, tools and techniques, prototyping and
debugging.
* Applications: systems programming, scientific computing, embedded
systems, graphics, security, model checking, compiler
generation, compiler optimization, decompilation.
Original results that bear on these and related topics are solicited.
Papers investigating novel uses and applications of program
manipulation are especially encouraged. Authors concerned about
the appropriateness of a topic are welcome to consult with the
program chair prior to submission.
SUBMISSION INFORMATION
Papers should be submitted electronically via the workshop's Web
page. Exceptionally, submissions may be emailed to the program
chair: asiapepm@comp.nus.edu.sg. Acceptable formats are PostScript
or PDF, viewable by gv. Submissions should not exceed 5000 words,
excluding bibliography and figures.
Submitted papers will be judged on the basis of significance,
relevance, correctness, originality, and clarity. They should clearly
identify what has been accomplished and why it is significant.
The work described should not have been previously published in
a major forum. Authors must indicate if a closely related paper
is also being considered for another conference or journal.
The proceeding of the symposium will be published by ACM Press.
A special issue of Higher-Order Symbolic Computation is also
planned.
LOCAL ARRANGEMENT
Mizuhito Ogawa (NTT, Japan)
GENERAL CHAIR
Kenichi Asai (Ochanomizu University, Japan)
PROGRAM CHAIR
Wei-Ngan Chin (National University of Singapore, Singapore)
PROGRAM COMMITTEE
Manuel Chakravarty (University of New South Wales, Australia)
Tyng-Ruey Chuang (Academia Sinica, Taiwan)
Charles Consel (ENSEIRB, France)
Oege de Moor (University of Oxford, UK)
Masami Hagiya (University of Tokyo, Japan)
Nevin Heintze (Agere Systems, USA)
Neil Jones (Univ of Copenhagen, Denmark)
Yanhong Annie Liu (SUNY at Stony Brook, USA)
Atsushi Ohori (JAIST, Japan)
Alberto Pettorossi (University of Roma, Italy)
Simon Peyton Jones (Microsoft, UK)
Carolyn Talcott (Stanford University, USA)
Zhe Yang (University of Pennsylvania, USA)
From rrosebru@mta.ca Wed Dec 12 16:32:23 2001 -0400
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Date: Thu, 6 Dec 2001 17:35:59 +0100
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Subject: categories: Positions in Paris 7 University
From: Pierre-Louis Curien
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Universite Paris 7 - CNRS
Laboratoire d'Informatique Algorithmique : Fondements et Applications
(LIAFA)
Laboratoire Preuves,, Programmes et Systemes (PPS)
APPEL A CANDIDATURES / ANNOUNCEMENT
Il y aura au concours 2002 / University Paris 7 will hire in 2002
- 2 postes de professeur d'universite / 2 professors
- 3 postes de maitres de conferences / 3 assistant professors
en informatique / in computer science.
La recherche en informatique est repartie sur deux laboratoires / There
are two computer science laboratories at Paris 7:
- LIAFA (algorithms and combinatorics, automata, modelisation and
verification)
- PPS (logic and programming)
Les deux laboratoires souhaitent renforcer et elargir leurs
thematiques / Both laboratories seek to reinforce and enlarge their
research themes.
Profils recherches / some possible research profiles for applicants
- tous les domaines de competence actuels des deux laboratoires / all
present themes of LIAFA and PPS
- LIAFA: bases de donnees, bio-informatique, cryptographie, ingenierie
de la
langue, systemes a evenements discrets / data bases, bio-informatics,
cryptography, computational linguistics, discrete event systems
- PPS: nous recherchons une ouverture sur les objets, ainsi que sur
concurrence et mobilite / we seek expertise on objects, and on mobility
and concurrency
Application information: The positions will be officially open for
application in early 2002. Only candidates who have gone through the
national Qualification procedure (application during the autumn of year
n for applying to a position in year n+1) are eligible. Rather fluent
knowledge of French is expected for teaching. The five positions are
permanent positions, starting october 2002.
Pour plus d'information sur les deux laboratoires / URL links of the two
labs
http://www.liafa.jussieu.fr
http://www.pps.jussieu.fr
Contact :
Daniel KROB - Directeur du LIAFA - dk@liafa.jussieu.fr
Pierre-Louis CURIEN - Directeur de PPS - curien@pps.jussieu.fr
From rrosebru@mta.ca Wed Dec 12 16:32:27 2001 -0400
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Date: Mon, 10 Dec 2001 21:07:44 -0800
From: Toby Bartels
To: categories@mta.ca
Subject: categories: Do catgory theorists like philosophy?
Message-ID: <20011210210743.A19228@math-cl-n03.ucr.edu>
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[note from moderator: please direct any responses to the poster, thanks]
It seems prima facie obvious that category theory ought to attract
mathematicians with a philosophical bent, given its potential to
revolutionise foundational issues and the perspective on the nature of
structure that it affords. But do category theorists interest themselves
in aspects of philosophy not directly related to mathematics? I'd be
interested to hear from the category theorists on this group, as well as
about the category theorists that you know or knew well enough to give a
definitive answer.
Of course, if ostensibly nonmathematical philosophy ended up having a
relation to mathematical philosophy or to category theory itself, that
counts too.
I hope that this somewhat extracurricular question is not considered out
of place, and welcome correction if it is.
-- Toby Bartels
toby@math.ucr.edu
From rrosebru@mta.ca Fri Dec 14 09:01:35 2001 -0400
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Date: Fri, 14 Dec 2001 12:29:49 +0100 (MET)
From: Jiri Adamek
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To: categories net
Subject: categories: infinite trees
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What is actually a first reference for the fact that for every polynomial
endofunctor of Set a final coalgebra consists of all properly labelled
(finite and infinite) trees? I suspect the first authors to study this
were Arbib and Manes in their
"Parametrized data types do not need...", Information and control 52
(1982), 139-158.
However, it is obvious from that paper that Arbib and Manes were
definitely unaware of the general statement, which explains why they only
mention some special cases in their book in 1986.
Jiri Adamek
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
alternative e-mail address (in case reply key does not work):
J.Adamek@tu-bs.de
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
From rrosebru@mta.ca Sun Dec 16 18:41:02 2001 -0400
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From: Peter Selinger
Message-Id: <200112162206.fBGM6PW27582@quasar.mathstat.uottawa.ca>
Subject: categories: Ottawa Logic Group invites graduate student applications
To: categories@mta.ca (Categories List)
Date: Sun, 16 Dec 2001 17:06:25 -0500 (EST)
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Dear colleagues,
could you please circulate this announcement to anyone who might be
interested, particularly bright prospective graduate students.
Thanks, -- Phil, Rick, and Peter
The University of Ottawa Logic Group (in the Mathematics Department) is
looking for qualified PhD and Master's students. Our group consisting
of Richard Blute, Philip Scott, and Peter Selinger, works in a wide
range of areas of Logic and theoretical computer science, including:
category theory, categorical logic, proof theory, linear logic, type
theory, programming language theory, theoretical computer science, and
foundations of physics.
The Mathematics Department Graduate Program is part of the
Ottawa-Carleton Institute of Mathematics and Statistics, and provides
a wide range of courses and programs. Ottawa is the capital of Canada,
and a beautiful, bilingual (English and French) city, 200 km west of
Montreal.
Funding for qualified PhD level students in logic is available--an
open competition for graduate students will be held in early February.
It is also possible to apply to pursue graduate studies with us
through SITE (the School of Information Technology and Software
Engineering--Computer Science Division), since some of us are
cross-appointed in Computer Science. Please enquire.
Outside the Mathematics Department, the Logic Group has several other
members, including Amy Felty, Tomoyuki Yamakami, and Luigi Logrippo
(SITE), Doug Howe and Leopoldo Bertossi (Carleton School of Computer
Science), and Mathieu Marion (Philosophy, Ottawa), plus visitors and
postdocs. We have a weekly seminar in all areas of logic and in which
graduate students are encouraged to present their work.
For more information, see:
http://quasar.mathstat.uottawa.ca/lfc/ (Logic Group)
http://quasar.mathstat.uottawa.ca/grad/ (Graduate Program in Math)
Preliminary applications can be completed online, at
http://quasar.mathstat.uottawa.ca/grad/apply.html
Or for further information, please contact us:
Philip Scott scpsg@matrix.cc.uottawa.ca
Richard Blute rblute@mathstat.uottawa.ca
Peter Selinger selinger@mathstat.uottawa.ca
From rrosebru@mta.ca Mon Dec 17 15:06:15 2001 -0400
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Date: Fri, 30 Nov 2001 18:19:31 -0800 (PST)
From: mjhealy@redwood.rt.cs.boeing.com (Michael Healy 425-865-3123)
Message-Id: <200112010219.SAA09113@lilith.rt.cs.boeing.com>
To: categories@mta.ca
Subject: categories: Change of email address
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I'm leaving industry to pursue applying category theory in my work---and,
incidentally, finally have time to really learn it! My email address from
now on is
mjhealy@u.washington.edu
Regards to all,
Mike
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Date: Mon, 17 Dec 2001 14:57:18 +0000
From: Ronnie Brown
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To: categories@mta.ca
Subject: preprint: Multiple categories: the equivalence of a globular anda cubical approach
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This is to advise that a revised version of
Title: Multiple categories: the equivalence of a globular and a cubical
approach
Authors: Fahd A.A. Al-Agl, Ronald Brown and Richard Steiner
as accepted today for Advances in Mathematics has been placed on the
xArchive at math.CT/0007009. There are some minor but useful corrections
and some new pictures.
--
Prof R. Brown,
School of Informatics, Mathematics Division,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT,
United Kingdom
Tel. direct:+44 1248 382474|office: 382681
fax: +44 1248 361429
World Wide Web: home page:
http://www.bangor.ac.uk/~mas010/
(Links to survey articles: Higher dimensional group theory
Groupoids and crossed objects in algebraic topology)
Raising Public Awareness of Mathematics CDRom Version 1.1
http://www.bangor.ac.uk/~mas010/CDadvert.html
Symbolic Sculpture and Mathematics:
http://www.cpm.informatics.bangor.ac.uk/sculmath/
Centre for the Popularisation of Mathematics
http://www.cpm.informatics.bangor.ac.uk/
From rrosebru@mta.ca Tue Dec 18 16:21:53 2001 -0400
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for categories-list@mta.ca; Tue, 18 Dec 2001 16:13:40 -0400
Date: Mon, 17 Dec 2001 09:04:59 -0500 (EST)
From: larry moss
To: cmcs@indiana.edu
Subject: categories: CFP: CMCS 02 2nd Call For Papers
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[Apologies for multiple copies]
SECOND CALL FOR PAPERS
CMCS2002
5th International Workshop on
Coalgebraic Methods in Computer Science
Grenoble, France
6-7 April 2002
A satellite workshop of ETAPS 2002
Aims and Scope
--------------
During the last few years, it is becoming increasingly clear that a great
variety of state-based dynamical systems, like transition systems,
automata, process calculi and class-based systems can be captured
uniformly as coalgebras. Coalgebra is developing into a field of its own
interest presenting a deep mathematical foundation, a growing field of
applications and interactions with various other fields such as reactive
and interactive system theory, object oriented and concurrent programming,
formal system specification, modal logic, dynamical systems, control
systems, category theory, algebra, analysis, etc. The aim of the workshop
is to bring together researchers with a common interest in the theory of
coalgebras and its applications.
The topics of the workshop include, but are not limited to:
the theory of coalgebras (including set theoretic and categorical
approaches);
coalgebras as computational and semantical models (for programming
languages, dynamical systems, etc.);
coalgebras in (functional, object-oriented, concurrent)
programming;
coalgebras and data types;
(coinductive) definition and proof principles for coalgebras (with
bisimulations or invariants);
coalgebras and algebras;
coalgebraic specification and verification;
coalgebras and (modal) logic;
coalgebra and control theory (notably of discrete event and hybrid
systems).
The workshop will provide an opportunity to present recent and ongoing
work, to meet colleagues, and to discuss new ideas and future trends.
Previous workshops of the same series have been organized in Lisbon,
Amsterdam, Berlin, and Genova. The proceedings appeared as Electronic
Notes in Theoretical Computer Science (ENTCS) Volumes 11,19, 33, and 41.
You can get an idea of the types of papers presented at the meeting by
looking at the tables of contents of the ENTCS volumes from the meetings,
available at the ENTCS page. For venue, registration and suggested
accommodation see the ETAPS2002 web page, http://www-etaps.imag.fr/
Submissions
-----------
Submissions will be evaluated by the Program Committee for inclusion in
the proceedings, which will be published in the ENTCS series. Papers must
contain original contributions, be clearly written, and include
appropriate reference to and comparison with related work. Papers (of at
most 15 pages) should be submitted electronically as uuencoded PostScript
files at the address cmcs@cs.indiana.edu. A separate message should also
be sent, with a text-only one-page abstract and with mailing addresses
(both postal and electronic), telephone number and fax number of the
corresponding author.
Important Dates
----------------
Deadline for submission: 8 January 2002.
Notification of acceptance: 20 February 2002.
Final version due: 10 March 2002.
Workshop dates: 6-7 April 2002.
Invited Speakers
----------------
Our list of invited speakers is coming, and will be
announced on the web page for the conference,
http://www.cs.indiana.edu/cmcs/
Program Committee
-------------------
J. Adamek (Braunschweig)
Alexandru Baltag (Amsterdam)
Jesse Hughes (Nijmegen)
H. Peter Gumm (Marburg)
Alexander Kurz (Amsterdam)
Bart Jacobs (Nijmegen)
Marina Lenisa (Udine)
Ugo Montanari (Pisa)
Larry Moss (chair, Bloomington, IN)
Ataru T. Nakagawa (Tokyo)
John Power (Edinburgh)
Horst Reichel (Dresden)
Jan Rutten (Amsterdam)
For more information
---------------------
http://www.cs.indiana.edu/cmcs/
cmcs@cs.indiana.edu
From rrosebru@mta.ca Tue Dec 18 16:35:29 2001 -0400
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Date: Mon, 17 Dec 2001 17:26:27 -0500 (EST)
From: LICS
Message-Id: <200112172226.fBHMQPu08113@moose.cs.indiana.edu>
To: categories@mta.ca
Subject: categories: job: faculty position in logic
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[Sincere apologies for duplicates]
Indiana University invites applications for a tenure-track
assistant professor position in applied logic.
Please see www.informatics.indiana.edu/positions/logic.html
for details. Applications received within the next few
weeks are likely to still get full consideration.
Applicants are welcome to email to foc@cs.indiana.edu
to notify of their mailed application, and to provide
pointers to any pertinent on-line documentation.
From rrosebru@mta.ca Tue Dec 18 19:36:02 2001 -0400
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for categories-list@mta.ca; Tue, 18 Dec 2001 19:33:56 -0400
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To: categories-list@mta.ca
From: "Valeria de Paiva"
To:
Subject: categories: CFP: IMLA 02 Call for Papers
Date: Mon, 17 Dec 2001 14:10:37 PST
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[... apologies for multiple copies ... ]
------------------- WORKSHOP ANNOUNCEMENT -----------------------
*** FIRST CALL FOR PAPERS ***
FLoC'2002 Workshop
IMLA 2: Intuitionistic Modal Logic and Applications '02
http://floc02.diku.dk/IMLA/
July 26, 2002
Copenhagen, Denmark
BACKGROUND
Constructive and modal logics are of foundational and practical relevance
to Computer Science. Constructive logics are used as type disciplines for
programming languages, as metalogics for denotational semantics, in the
paradigm of program extraction from proofs and for interactive proof
development in automated deduction systems such as Agda, Coq, Twelf,
Isabelle, HOL, NuPrl and Plastic. Modal logics like temporal logics, dynami=
c
logics and process logics are used in industrial-strength applications as
concise formalisms for capturing reactive behaviour.
Although constructive and modal frameworks have typically been investigated
separately, a growing body of published work shows that both paradigms can
(and should) be fruitfully combined. The goal of this workshop is to
stimulate more systematic study of constructive or Intuitionistic Modal
Logics and, in parallel of modal type theories. It aims to
1. bring together two largely parallel communities - computer scientists
with a focus on proof theory and lambda calculi, and logicians and
philosphers with a focus on model theory;
2. bring together theoretically-oriented and the application-oriented
approaches, in the hope of productive interaction.
Theoretical / methodological issues centre around the question of how the
proof-theoretic strengths of constructive logics can best be combined with
the model-theoretic strengths of modal logics. Two basic questions are thus
"what is the right notion of proof?" and "what is the right way of making a
given modal logic constructive?".
Topics of interest for papers in the Workshop include, but are not limited
to:
* applications of intuitionistic necessity or possibility, strong
monads, or evaluation modalities,
* use of modal type theory to formalize mechanisms of abstraction
and refinement,
* applications of constructive modal logic and modal type theory to
formal verification, abstract interpretation, and program analysis and
optimization
* applications of modal types to integration of inductive and
co-inductive types, higher-order abstract syntax, strong functional
programming
* computational aspects of the Curry-Howard correspondence between
lambda calculi and logics
* extensions of this correspondence by other modalities or
quantifiers
* models of constructive modal logics such as algebraic, categorical=
,
Kripke, topological, realizability interpretations
* notions of proof for constructive modal logics
* extraction of constraints or programs, nonstandard information
extraction techniques
* proof search in constructive modal logic and implementations of it
FORMAT
The workshop will be an informal one-day meeting with two invited
talks, regular paper presentations, and discussion.
INVITED SPEAKERS
Giovanni Sambin (Padova, Italy)
Dana Scott (Pittsburgh, USA)
PUBLICATION
Workshop contributions must be original work that has not yet appeared
elsewhere. If accepted, the authors are expected to present their paper at
the workshop.
Workshop papers will be made available on the workshop web page and
will appear as a technical report handed out to all workshop
participants.
Authors of accepted papers will be invited to submit
full and revised versions of the Workshop papers to a special issue of
the Journal of Logic and Computation, for which there will be a second
round of refereeing.
SUBMISSIONS
All submissions should be single column, use 11 point font, and be at most
15 pages in length, preferably using the LaTeX llnc style. Papers should no=
t
be already published and should not be submitted for simultaneous
publication at another conference or workshop. Either send a .ps or .pdf
file to
M.Mendler@dcs.shef.ac.uk
or post a hard copy to
Dr Michael Mendler
The Department of Computer Science
Regent Court
211 Portobello Street
Sheffield
S1 4DP
UNITED KINGDOM
by the due date.
IMPORTANT DATES
IMLA submission deadline: April 5, 2002
IMLA notification : May 23, 2002
IMLA final version : June 20, 2002
PROGRAMME COMMITTEE
Natasha Alechina (Nottingham, UK)
Sergei Artemov (Cornell, USA)
Johan van Benthem (Amsterdam and Stanford)
Rajeev Gor=E9 (ANU, Australia)
Jean Goubault-Larrecq (ENS-Cachan, France)
Michael Mendler (Sheffield ,UK)
Eugenio Moggi (Genova, Italy)
Valeria de Paiva (Xerox PARC, USA)
Frank Pfenning (CMU, USA)
Carsten Schuermann (Yale, USA)
Alex Simpson (Edinburgh, UK)
ORGANISERS
Rajeev Gore (ANU, Australia)
Michael Mendler (Sheffield, UK)
Valeria de Paiva ( PARC, USA)
Workshop webpage: http://floc02.diku.dk/IMLA/
From rrosebru@mta.ca Thu Dec 20 13:15:06 2001 -0400
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Message-ID: <20011217221100.25488.qmail@web12203.mail.yahoo.com>
Date: Mon, 17 Dec 2001 14:11:00 -0800 (PST)
From: Galchin Vasili
Subject: categories: multigaphs/categories and constructivism
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Hello,
This is another philosophy of math question:
1) What is the constructivist position on infinite
multigraphs with loops?
2) What is the constructivist position on
infinite categories?
Regards, Bill Halchin
From rrosebru@mta.ca Thu Dec 20 13:15:09 2001 -0400
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Message-Id:
To: categories-list@mta.ca
Date: Mon, 17 Dec 2001 21:45:57 -0800 (PST)
From: Posina Venkata Rayudu
Subject: categories: perception to knowledge
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Hi,
Recently Prof. Lawvere suggested a model for how
experimental data lead to theory (see enclosed
posting). In this context I would like to mention that
there is similar problem in cognitive science. How do
we acquire knowledge or how is perceptual experience
transformed into conceptual knowledge? Moreover the
discussion on generals & particulars readily lends
itself to interpretation in cognitive terminology. We
may compare particulars with sensation, abstract
generals with categories (not the mathematical
category) such as =91dog=92, and concrete generals with
prototypes. Stretching little further, presentation of
abstract general can be compared with percept (of
dog).
Of all the disciplines to which category theory is
applied such as physics or computer science, I think
the most natural domain of applications for category
theory is cognitive science. Let me explain. Category
theory captures mathematical practice. In the domain
of mathematics, category theory provides a
mathematical account of the process of transforming
ignorance into knowledge. It is reasonable to treat
the growth of mathematical knowledge as a particular
case of knowing in general or cognition. Given that
category theory models a particular case of cognition
(mathematics), one strategy is to generalize the
category theoretic description of the particular case
of mathematical knowing to knowing in general or
cognition. The ease with which we can implement and
realize this research program is inversely
proportional to the =91distance=92 between mathematical
practice and cognitive processes. The more the
cognition is similar to mathematics, the less are the
changes or effort we need to make to the category
theoretic model of mathematics for it to accommodate
cognition in general. In view of the close resemblance
between the cognitive process and the pursuit of
mathematical knowledge (e.g. both describe real in
terms of imaginary), category theoretic study of
cognition is likely to be extremely fruitful.
One could motivate category theory in more concrete
terms. For example, the problem of how perceptual
experiences give rise to conceptual knowledge has a
facet to which category theory has already provided
solution. How do we go from figural or picturesque
perception (geometry, topology) to propositional or
symbolic thinking (algebra, logic)? Category theory
explicated the connections between logic and topology
and these insights can be brought to bear on
comparable perception-thought transformations, and it
possibly gives a cue to the role of thinking vis-=E0-vis
perception. =20
I am sorry if I said something really stupid about
category theory. I simply want to attract category
theorists to cognitive science.
Thanking you,
Sincerely,
Posina Venkata Rayudu
=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=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D
From: "F. William Lawvere" =20
To: categories@mta.ca=20
Subject: categories: Re: Sketches and Platonic Ideas=20
Date: Wed, 05 Dec 2001 04:36:21 =20
In my 1972 Perugia Notes I had made an attempt to
characterize the relation between these sorts of
mathematical considerations and philosophy by saying
that while platonism is wrong on the relation between
Thinking and Being, something analogous is correct
WITHIN the realm of Thinking. The relevant dialectic
there is between abstract general and concrete
general. Not concrete particular ("concrete" here does
not mean "real"). There is another crucial dialectic
making particulars (neither abstract nor concrete)
give rise to an abstract general; since experiments do
not mechanically give rise to theory, it is harder to
give a purely mathematical outline of how that
dialectic works, though it certainly does work. A
mathematical model of it can be based on the
hypothesis that a given set of particulars is somehow
itself a category (or graph), i.e., that the
appropriate ways of comparing the particulars are
given but that their essence is not. Then their
"natural structure" (analogous to cohomology
operations) is an abstract general and the
corresponding concrete general receives a
Fourier-Gelfand-Dirac functor from the original
particulars. That functor is usually not full because
the real particulars are infinitely deep and the
natural structure is computed with respect to some
limited doctrine; the doctrine can be varied, or
"screwed up or down" as James Clerk Maxwell put it, in
order to see various phenomena.
=3D=3D=3D=3D=3D
Posina Venkata Rayudu
C/o: Sri. S. S. Chalam
Advocate & Notary Public
H.No: 39-4-10, Innespeta
Rajahmundry =96 533102
Andhra Pradesh, India
Phone: 91 (0883) 444232
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Date: Wed, 19 Dec 2001 04:09:39 -0800 (PST)
From: Posina Venkata Rayudu
Subject: categories: language and thinking
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Hi,
Does the debate -elements and their belongingness vs.
functions and their composition- support Sapir-Whorf
hypothesis that the way we think is a function of the
language we use. In other words, language can
transform thinking. According to this doctrine of
linguistic relativity, =93users of markedly different
grammars are pointed by their grammars toward
different types of observations=85and hence are not
equivalent as observers, but must arrive at somewhat
different views of the world=94 (Whorf 1956, p. 221).
Whorf, B. L. (1956) Language, Thought, and Reality:
Selected Writings of Benjamin Lee Whorf (ed. J. B.
Carroll) MIT Press, Cambridge, MA.
Thanking you,
Sincerely,
Posina Venkata Rayudu
=3D=3D=3D=3D=3D
Posina Venkata Rayudu
C/o: Sri. S. S. Chalam
Advocate & Notary Public
H.No: 39-4-10, Innespeta
Rajahmundry =96 533102
Andhra Pradesh, India
Phone: 91 (0883) 444232
From rrosebru@mta.ca Thu Dec 20 13:10:17 2001 -0400
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X-Organisation: Faculty of Science, University of Amsterdam, The Netherlands
X-URL: http://www.science.uva.nl/
Date: Wed, 19 Dec 2001 13:58:40 +0100
From: Methods for Modalities
To: Methods for Modalities
Subject: categories: CFP: HyLo@LICS
Message-ID: <20011219135840.A6168@science.uva.nl>
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HyLo@LICS
4th WORKSHOP ON HYBRID LOGICS
LICS 2002 Affiliated Workshop
>>> JULY 25, 2002 <<<
Copenhagen, Denmark
FIRST CALL FOR PAPERS
THEME:
Hybrid logic is a branch of modal logic in which it is possible to
directly refer to worlds/times/states or whatever the elements of the
(Kripke) model are meant to represent. Although they date back to the
late 1960s, and have been sporadically investigated ever since, it is
only in the 1990s that work on them really got into its stride.
It is easy to justify interest in hybrid logic on applied grounds,
with the usefulness of the additional expressive power. For example,
when reasoning about time one often wants to build up a series of
assertions about what happens at a particular instant, and standard
modal formalisms do not allow this. What is less obvious is that the
route hybrid logic takes to overcome this problem (the basic
mechanism being to add nominals --- atomic symbols true at a unique
point --- together with extra modalities to exploit them) often
actually improves the behavior of the underlying modal formalism. For
example, it becomes far simpler to formulate modal tableau and
resolution in hybrid logic, and completeness and interpolation
results can be proved of a generality that is simply not available in
modal logic. That is, hybridization --- adding nominals and related
apparatus --- seems a fairly reliable way of curing many known
weaknesses in modal logic. For more general background on hybrid
logic, and many of the key papers, see the Hybrid Logics homepage:
http://www.hylo.net
HyLo@LICS is likely to be relevant to a wide range of people,
including those interested in description logic, feature logic,
applied modal logics, temporal logic, and labelled deduction.
Moreover, if you have an interest in the work of the late Arthur
Prior, note that this workshop is devoted to exploring ideas he first
introduced 30 years ago --- it will be an ideal opportunity to see
how his ideas have been developed in the intervening period.
In this workshop we hope to bring together researchers from all the
different fields just mentioned (and hopefully some others) in an
attempt to explore what they all have (and do not have) in common. If
you're unsure whether your work is of relevance to the workshop,
please check out the Hybrid Logics homepage. And do not hesitate to
contact the workshop organisers for more information. We'd be
delighted to tell you more. Contact details are give below.
SUBMISSIONS:
We invite the contribution of research papers to the workshop. Please
send electronically an extended abstract of up to 10 A4 size pages,
in PostScript format to: carlos@science.uva.nl BEFORE the 26st of
APRIL, 2002. Please note that all workshop contributors are required
by the LICS organizers to register for FLoC 2002.
IMPORTANT DATES:
Deadline for Submissions: April 26th, 2002
Notification of Acceptance: May 24th, 2002
Deadline for Final Versions: June 25th, 2002
CONTACT DETAILS:
Please visit http://www.hylo.net for further information.
Send all correspondence regarding the workshop to the organizers:
Carlos Areces
e-mail: carlos@wins.uva.nl
http://www.illc.uva.nl/~carlos
Patrick Blackburn
e-mail: patrick@coli.uni-sb.de
http://www.coli.uni-sb.de/~patrick
Maarten Marx
e-mail: marx@science.uva.nl
http://www.illc.uva.nl/~marx
Ulrike Sattler
e-mail: sattler@cs.rwth-aachen.de
http://www-lti.informatik.rwth-aachen.de/ti/uli-en.html
From rrosebru@mta.ca Thu Dec 20 13:10:19 2001 -0400
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for categories-list@mta.ca; Thu, 20 Dec 2001 13:05:57 -0400
Message-Id:
To: categories-list@mta.ca
Subject: categories: CFP: ICALP2002
From: icalp2002@lcc.uma.es
To: categories@mta.ca
Message-Id:
Date: Wed, 19 Dec 2001 14:39:55 +0100
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We apologize for possible multiple postings.
In http://www.lcc.uma.es/icalp2002 you can find a pdf
version of this call for paper.
+++++++++++++++++++++++++++++++++++++
Last information:
Second version of the WEB pages at http://www.lcc.uma.es/icalp2002
Workshops confirmed:
Computability and Complexity in Analysis (CCA 2002)
Algorithmic Methods and Models for Optimization of Railways ATMOS 2002
7 th International Workshop on Formal Methods for Industrial Critical Sys=
tems
Foundations of Wide Area Network Computing
Invited speakers confirmed:
Heikki Mannila
Manuel Hermenegildo
+++++++++++++++++++++++++++++++++++++
Call for Papers
ICALP 2002=20
29th International Colloquium on=20
Automata, Languages and Programming=20
July 8-13, 2002, M=E1laga, Spain=20
Camera Ready: April 16, 2002
The 29th annual meeting of the European Association of Theoretical
Computer Science will be held in M=E1laga, Spain, July 8-13, 2002 (at the
E.T.S. Ingenier=EDa Inform=E1tica).=20
As with the Journal Theoretical Computer Science (TCS), the scientific
program of the Colloquium will be split into two parts: Track A of the
meeting will correspond to Algorithms, Automata, Complexity and Games,
while Track B to Logic, Semantics and Theory of Programming.=20
SUBMISIONS: Authors are invited to submit extended abstract of their
papers, presenting original contributions to the theory of computer
science. Detailed instructions for paper submissions will be found on
the conference webpage (http://www.lcc.uma.es/icalp2002) and in future
calls for papers. Submissions should consist of: a cover page, with the
author's full name, address, fax number, e-mail address, a 100-word
abstract, keywords and to which track (A or B) the paper is being
submitted and an extended abstract describing original research. At
least one author of an accepted paper should be available to present it
at the conference. Simultaneous submission to other conferences with
published proceedings is not allowed.=20
Further information (dates and instructions for submissions, workshops,
registration, location and travel) will be provided in future
announcements.=20
ORGANIZING COMMITEE: Buenaventura Clares (University of Granada),
Ricardo Conejo (University of M=E1laga), Inmaculada Fortes (University of
M=E1laga), Llanos Mora (University of M=E1laga), Rafael Morales (co-Chair,
University of M=E1laga), Marlon Nu=F1ez (University of M=E1laga), Jos=E9 Lu=
is
P=E9rez de la Cruz (University of M=E1laga), Gonzalo Ramos (University of
M=E1laga), Francisco Triguero (co-Chair, University of M=E1laga), Jos=E9 Lu=
is
Trivi=F1o (University of M=E1laga).=20
IMPORTANT DATES:=20
Workshops proposal: November 8, 2001=20
Submissions: January 14, 2002=20
Notification: March 20, 2002=20
CONFERENCE CO-CHAIRS=20
Prof. Francisco Triguero
Prof. Rafael Morales=20
Universidad de M=E1laga=20
E.T.S. Ingenier=EDa Inform=E1tica=20
Dept. Lenguajes y Ciencias de la Computaci=F3n=20
Bulevar Louis Pasteur, 35=20
29071 - M=E1laga (SPAIN)=20
e-mail: icalp2002@informatica.uma.es=20
PROGRAM COMMITEE
Track A=20
Ricardo Baeza-Yates (U. Chile)=20
Volker Diekert (U. Stuttgart)=20
Paolo Ferragina (U. Pisa)=20
Catherine Greenhill (U. Melbourne)=20
Torben Hagerup (U. Frankfurt)=20
Johan H=E5stad (KTH, Stockholm)=20
Gabriel Istrate (Los Alamos)=20
Claire Kenyon (U. Paris XI)=20
Der-Tsai Lee (Acad. Sinica, Taipei)=20
Heikki Mannila (Nokia, Helsinki)=20
Elvira Mayordomo (U. Zaragoza)=20
Helmut Prodinger (U. Witwatersrand, South Africa)=20
Jan van Leeuwen(U. Utrecht)=20
Paul Vit=E1nyi (CWI, Amsterdam)=20
Peter Widmayer (ETH Z=FCrich) (Chair)=20
Gerhard Woeginger (T.U. Graz)=20
Christos Zaroliagis (U. Patras)=20
Track B=20
Mart=EDn Abadi (Bell Labs Research, Lucent)=20
Roberto Amadio (U. Provence)=20
Gilles Barthe (INRIA-SophiaAntipolis)=20
Manfred Droste (University of Technology Dresden)=20
C=E9dric Fournet (Microsoft Cambridge)=20
Matthew Hennessy (U. Sussex) (Chair)=20
Furio Honsell (U. Udine)=20
Peter O'Hearn (Queen Mary & W. C. London)=20
Fernando Orejas (U.P.Catalunya)=20
Ernesto Pimentel (U. M=E1laga)=20
David Sands (Chalmers University of Technology and G=F6teborg University)=
=20
Dave Schmidt (U. Kansas)=20
Gheorghe Stefanescu (U. Bucharest)=20
Vasco Vasconcelos (U. Lisbon)=20
Thomas Wilke (U. Kiel)
+++++++++++++++++++++++++++++++++++++++++++
Malaga University uses Christmas holidays for backup and maintenance of his=
network.
If you have problem to arrive to ICALP 2002 main page, please retry again l=
ater.
If you get to the page http://www.lcc.uma.es but not to the page http://www=
=2Elcc.uma.es/icalp2002 contact with us:
conejo@lcc.uma.es
morales@lcc.uma.es
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Date: Thu, 20 Dec 2001 15:17:01 -0500
To: cat-dist@mta.ca
From: Charles Wells
Subject: categories: Re: language and thinking
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Learning category theory (which I did after I wrote a dissertation and
several papers in finite fields) certainly changed and improved the way I
did mathematics. The change indeed deserves to be called a
transformation. In a similar way, my ability to program in Basic improved
remarkably when I learned the basic ideas of structured programming and
did a little programming in Pascal. But I am reasonably sure that these
transformations in my thinking occurred because I learned important new
concepts such as limit, adjoint, while-loop, etc. Learning new concepts
transforms one's thinking. I am not a linguist, but I know something of
Whorf's ideas; I don't understand how one can disentangle the effect of
knowing the different concepts that different cultures have from the
effect of knowing their language.
This brings up the question: Can concepts be differentiated from language?
I say via introspection that the answer is "certainly", because when I
concentrate on a mathematical problem (or how to reassemble a machine or
write a complicated program) the "talking" in my head goes away and is
replaced by pictorial concepts located in mental space. Some people claim
that this never happens to them. If that is true, it would appear that
people come in two different varieties, from Mars and from Venus maybe.
But I suspect that the people who claim it never happens are simply wrong:
they lack sufficient introspective ability.
--Charles Wells
>Does the debate -elements and their belongingness vs.
>functions and their composition- support Sapir-Whorf
>hypothesis that the way we think is a function of the
>language we use. In other words, language can
>transform thinking. According to this doctrine of
>linguistic relativity, =93users of markedly different
>grammars are pointed by their grammars toward
>different types of observations=85and hence are not
>equivalent as observers, but must arrive at somewhat
>different views of the world=94 (Whorf 1956, p. 221).
>
>Whorf, B. L. (1956) Language, Thought, and Reality:
>Selected Writings of Benjamin Lee Whorf (ed. J. B.
>Carroll) MIT Press, Cambridge, MA.
>
>Thanking you,
>Sincerely,
>Posina Venkata Rayudu
>
>=3D=3D=3D=3D=3D
>Posina Venkata Rayudu
>C/o: Sri. S. S. Chalam
>Advocate & Notary Public
>H.No: 39-4-10, Innespeta
>Rajahmundry =96 533102
>Andhra Pradesh, India
>Phone: 91 (0883) 444232
Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
Send all mail to:
105 South Cedar St., Oberlin, Ohio 44074, USA.
email: charles@freude.com.
home phone: 440 774 1926.
professional website: http://www.cwru.edu/artsci/math/wells/home.html
personal website: http://www.oberlin.net/~cwells/index.html
genealogical website:
http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/
NE Ohio Sacred Harp website: http://www.oberlin.net/~cwells/sh.htm
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From: S.J.Vickers@open.ac.uk
To: categories@mta.ca
Subject: categories: Re: language and thinking
Date: Fri, 21 Dec 2001 10:05:48 -0000
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> This brings up the question: Can concepts be differentiated from language?
Two quite different examples: first, a very practical one.
This is of vital importance in teaching computing. Programming languages
come and go,
and to be reasonably future-proof a programming course must go beyond merely
"teaching a programming language" and bring out concepts.
Some evidence that concepts can be differentiated from language is seen in
the graphical Integrated Development Environments (IDEs) for developing
object oriented programs. For instance, if you compare those for Java with
those for C++ you find that broadly similar diagrammatic metaphors (grab an
object, place it somewhere, link it to other objects to handle certain
events, etc.) get implemented in rather different ways in different
languages.
This sounds very like Charles's replacement of talking in his head by
pictorial concepts. Its effectiveness in IDEs is indisputable, and it seems
to be because the language by itself in some way hobbles your thought
processes (cf. Basic programming improved by knowing Pascal).
A quite different example is that of foundations, how choice of logic
affects what you can recognize as mathematics.
But don't get me going on that.
Merry Christmas,
Steve Vickers.
From rrosebru@mta.ca Sat Dec 22 10:41:00 2001 -0400
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Date: Fri, 21 Dec 2001 16:38:35 +0100 (MET)
From: Christophe Ringeissen
To: categories@mta.ca
Subject: categories: CFP: AMAST'2002
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[We apologize if you receive multiple copies of this message]
AMAST 2002 CALL FOR PAPERS
9-th International Conference on Algebraic Methodology And Software Technology
AMAST 2002, September 9-13, 2002
St. Gilles les Bains, Reunion Island, France
* Important Dates:
Paper submissions February 1, 2002
Notification of paper acceptance April 27, 2002
Camera ready papers June 1, 2002
AMAST 2002 conference September 9-13, 2002
* Topics:
As in previous years, we invite papers reporting original research
on setting software technology on a firm mathematical basis.
Of particular interest is research on using algebraic,
logic, and other formalisms suitable as foundations for software
technology, as well as software technologies developed by means of
logic and algebraic methodologies.
* Submissions:
We invite prospective authors to submit electronically previously
unpublished papers of high quality. Papers must be no longer than 15
pages (6 pages for system demonstrations) and should be prepared using
LaTeX and the LNCS style that can be downloaded from the URL:
http://www.springer.de/comp/lncs/authors.html
Please send a fully self-contained PostScript file to
amast@loria.fr
As in the past, the AMAST'2002 proceedings will be published by
Springer-Verlag in the Lecture Notes in Computer Science Series.
* Program Committee:
V.S. Alagar, E. Astesiano, M. Bidoit, D. Bolignano,
M. Broy, J. Fiadeiro, B. Fischer, K. Futatsugi, A. Haeberer,
N. Halbwachs, A. Haxthausen, D. Hutter, P. Inverardi, B. Jacobs,
M. Johnson, H. Kirchner (PC chair), P. Klint, T. Maibaum, Z. Manna,
J. Millen, P. Mosses, F. Orejas, R. de Queiroz, T. Rus,
C. Ringeissen (PC chair assistant), D. Sannella, P.-Y. Schobbens,
G. Scollo, A. Tarlecki, M. Wirsing
* Local Organization Chair: Teodor Knapik, Univ. de la Reunion
* Further information:
For regularly updated details of the conference
organization send email to amast@loria.fr
or visit the AMAST'2002 web page:
http://www.loria.fr/conferences/amast2002
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Date: Wed, 26 Dec 2001 09:06:43 -0500 (EST)
From: Peter Freyd
Message-Id: <200112261406.fBQE6hH19209@saul.cis.upenn.edu>
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In the film "A Beautiful Mind," the first indication that John Nash is
regaining his abilities is a conversation with a student ("Galois
extensions are really the same as covering spaces!" exclaims Toby, the
student). The scene ends with Nash and the student bent over some
papers and -- as I heard it -- the student says:
"Functor...Two...Categories"
(The film's not bad but its relation to Nash's life is tangential. If I
had not absorbed a hint from a review, I think I might have stomped
out in the middle because of the ridiculous portrayal of what
mathematicians do. It turns out that the filmmakers are playing a game
-- very effective with most of the audience -- in sliding between
perceptions: ours and Nash's.)
From rrosebru@mta.ca Sat Dec 29 20:19:27 2001 -0400
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for categories-list@mta.ca; Sat, 29 Dec 2001 20:11:34 -0400
To: categories@mta.ca
Subject: categories: Enriched locally presentable categories
From: Mark Hovey
Date: 26 Dec 2001 07:18:19 -0500
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I am still trying to understand some enriched category theory. Suppose
V is a closed symmetric monoidal category that is also locally
presentable. Suppose C is a small V-category. I am interested in the
category of V-functors from C to V, and, in particular, I want to know
that it is locally presentable. Might need some hypotheses on C for
this, but I would prefer to avoid hypotheses on the actual functors.
This time I have actually looked in Kelly's book and I did not see it,
but I confess to finding this subject rough going so might have missed
it. On the other hand, my library is closed for the holiday, so I have
not looked at Adamek and Rosicky's book on enriched category theory yet.
I guess the generators ought to be the representable functors. I know
everything is a weighted colimit of representables, but I don't know
whether this colimit is filtered enough, nor do I know whether one can
get away with weighted colimits instead of ordinary ones.
One direction this might go is to develop a theory of locally
presentable in an enriched sense, using weighted colimits instead of
colimits. I would prefer to avoid that if possible.
Happy holidays to all.
Mark Hovey
From rrosebru@mta.ca Mon Dec 31 10:55:38 2001 -0400
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for categories-list@mta.ca; Mon, 31 Dec 2001 10:49:42 -0400
Date: Mon, 31 Dec 2001 19:30:27 +1100 (EST)
From: maxk@maths.usyd.edu.au (Max Kelly)
Message-Id: <200112310830.fBV8URA305918@milan.maths.usyd.edu.au>
To: categories@mta.ca
Subject: categories: Re: Enriched locally presentable categories
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Mark Hovey's letter of 26 Dec - known as Boxing Day in the British-speaking
world - suggests that it might be a good idea to develop a theory of local
presentability and all that in the context of enriched categories. In fact
such a theory was developed in my paper [Structures defined by finite limits
in the enriched context I, Cahiers de Top. et Geom. Differentielles 23 (1982),
3 - 42]. Everything works very smoothly; but there are a few annoying
misprints, many of which seem to be my own fault. Further developments can be
found in [Blackwell-Kelly-Power, Two-dimensional monad theory, J. Pure Appl.
Algebra 59 (1989), 1 - 41] and in [Kelly-Power, Adjunctions whose counits are
coequalizers and presentations of finitary enriched monads, J. Pure Appl.
Algebra 89 (1993), 163 - 179], among other papers of myself and of others;
Brian Day, Steve Lack, John Power, and Ross Street have all written on related
matters.
Please accept, Mark, my best wishes for your future work in this direction. In
any case, New Year's Eve is a fine time to send greetings more generally to
Bob and all on this Bulletin Board.
Warm regards - Max Kelly.
From rrosebru@mta.ca Mon Dec 31 10:55:41 2001 -0400
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for categories-list@mta.ca; Mon, 31 Dec 2001 10:51:50 -0400
Message-ID: <000b01c191d5$8cc90390$0e034ed4@WALTER>
From: "RFC Walters"
To:
Subject: categories: New address: RFC Walters
Date: Mon, 31 Dec 2001 09:31:29 +0100
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New email address
robert.walters@uninsubria.it
New mail address
Universita' dell'Insubria
Dipartimento di Scienze CC, FF e MM
Via Valleggio 11
22100 COMO, Italy
Web homepage
http://www.unico.it/~walters/
My recent papers are available there: the latest being
P. Katis, N. Sabadini, R.F.C. Walters
Feedback, trace and fixed point semantics
We introduce a notion of category with feedback-with-delay, closely
related to the notion of traced monoidal category, and show that the
Circ construction of [JPAA 115: 141--178, 1997] is the free category
with feedback on a symmetric monoidal category. Combining with the Int
construction of Joyal-Street-Verity [Math. Proc. Camb.Phil. Soc., 119,
447-468, 1996] we obtain a description of the free compact closed
category on a symmetric monoidal category. We thus obtain a categorical
analogue of the classical localization of a ring with respect to a
multiplicative subset. In this context we define a notion of fixed-point
semantics of a category with feedback which is seen to include a variety
of classical semantics in computer science.
Others:
R. Rosebrugh, N. Sabadini, R. F. C. Walters, Minimization and Minimal
Realization in Span(Graph), submitted
P. Katis, N. Sabadini, RFC Walters, Classes of finite state automata for
which compositional minimization is linear time
Fabio Gadducci, Piergiulio Katis, Ugo Montanari, Nicoletta Sabadini,
Robert F.C. Walters, Comparing cospan-spans and tiles via a Hoare-style
process calculus
From rrosebru@mta.ca Mon Dec 31 10:55:41 2001 -0400
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for categories-list@mta.ca; Mon, 31 Dec 2001 10:53:25 -0400
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Date: Mon, 31 Dec 2001 09:21:38 -0500 (EST)
From: Michael Barr
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To: Categories list
Subject: categories: Natural disasters
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It seems like there are natural disasters, first in Sydney and, to a
lesser extent, in Buffalo (82 inches of snow by one report) and I would
like to enquire how all are.
Michael
From rrosebru@mta.ca Mon Dec 31 10:55:44 2001 -0400
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for categories-list@mta.ca; Mon, 31 Dec 2001 10:52:59 -0400
Date: Mon, 31 Dec 2001 19:55:56 +1100 (EST)
From: maxk@maths.usyd.edu.au (Max Kelly)
Message-Id: <200112310855.fBV8tun306897@milan.maths.usyd.edu.au>
To: categories@mta.ca
Subject: categories: Re: Two categories or 2-categories?
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I don't know quite what Peter's point is here: there is no difference between
the spoken forms of "2-categories" and "two-categories". I think we all write
"2-categories", as we write "n-categories" and "w-categories", where I am
making-do with "w" for a lower-case Greek omega. Yet Blackwell, Power, and I,
when we considered general questions about the algebras for 2-monads and the
various kinds of strict and non-strict morphisms of these and some adjunctions
between the 2-categories that arise, entitled our paper "Two-dimensional monad
theory". I don't think "2-monad theory" would have represented our concerns as
well, being capable of interpretation as meaning a wider study than ours, or a
narrower one, depending on how it was taken by the reader. To the Australian
Research Council, such work is described as research on two-dimensional
universal algebra.
What do others think?
Regards - Max.
From rrosebru@mta.ca Mon Dec 31 21:02:41 2001 -0400
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for categories-list@mta.ca; Mon, 31 Dec 2001 20:54:53 -0400
Subject: categories: Re: Natural disasters
From: Duraid Madina
To: Categories list
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The fires here in Sydney are still blazing away, but that didn't stop
the new year's fireworks display from going ahead just a few hours ago.
As far as I am aware, no (human) lives have been lost, but there's been
a great deal of property damage, and an immense amount of national
parkland has been burned away.
In truth, most Sydneysiders are unaffected except for the appalling air
quality and the fact that a fine layer of ash has been deposited over
the entire city, which people will no doubt discover on their desks as
they return to work!
You can take a look at some eerie pictures of the Sydney skyline here:
http://www.smh.com.au/news/0112/26/gallery/snapshot1.html
Duraid
On Tue, 2002-01-01 at 02:21, Michael Barr wrote:
> It seems like there are natural disasters, first in Sydney and, to a
> lesser extent, in Buffalo (82 inches of snow by one report) and I would
> like to enquire how all are.
>
> Michael
>
From rrosebru@mta.ca Mon Dec 31 21:02:44 2001 -0400
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From: baez@math.ucr.edu
Message-Id: <200112311647.fBVGlLL06385@math-cl-n05.ucr.edu>
Subject: categories: Two categories or 2-categories
To: categories@mta.ca (categories)
Date: Mon, 31 Dec 2001 08:47:21 -0800 (PST)
In-Reply-To: <200112310855.fBV8tun306897@milan.maths.usyd.edu.au> from "Max Kelly" at Dec 31, 2001 07:55:56 PM
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Max Kelly writes:
> I don't know quite what Peter's point is here: there is no difference
> between the spoken forms of "2-categories" and "two-categories".
I forget who said what, but I think the issue was that when folks
are talking in this movie, you can't easily tell whether they are
saying "2-categories" or "two categories", i.e. a couple of categories.
This problem comes up a lot in my life, and I am glad to see it
finally showing up in a major motion picture! E.g., I must be
careful never to say "functor between two categories", replacing
it by "functor from one category to another".
I would be shocked if they were talking about 2-categories in
this movie. Even mentioning categories must seriously diminish
their ticket sales, much less 2-categories.
On a wholly different note, how are the category theorists in
Sydney?