Date: Wed, 2 Feb 1994 23:22:48 +0400 (GMT+4:00)
Subject: Re: question on homotopies
Date: Tue, 1 Feb 1994 15:05:25 -0500
From: James Stasheff
> From: Michael Barr
>
> In the category of chain complexes (over some abelian category)
> it is clear that if you identify homotopic arrows, you will also
> invert homotopy equivalences. Does anyone know if the converse is
> true? If you invert homotopy equivalences, do you wind up identifying
> homotopic arrows? What if you replace homotopy by homology? Does
> inverting maps that induce isomorphism on homology have the effect of
> identifying maps that induce the same map on homology?
>
> Michael
>
In the category of chain algebras, this has been well studied
the point being that the inverse (up to homotpy) of a
multiplicative homotopy equivalence is generally NOT
mutiplicative
the new category is known as
DASH
and has an extensive literature
jim
Date: Wed, 2 Feb 1994 23:26:12 +0400 (GMT+4:00)
Subject: Is *Loc* regular?
Date: Wed, 2 Feb 1994 11:21:42 +0100
From: Todd Wilson
I have been wondering lately whether every regular monomorphism of the
category of frames is universal (= pushout stable). But it is not hard to
see (using, e.g., Kelly's "Monomorphisms, epimorphisms, and pull-backs"
J.Austr.Math.Soc. 9, 1969) that this is equivalent to the question of
whether the category of locales is a regular category. Does anyone have an
answer to this question?
--Todd Wilson
Date: Fri, 4 Feb 1994 23:40:24 +0400 (GMT+4:00)
Subject: categories of graphs
Date: Fri, 04 Feb 1994 14:24:26 +0000
From: "Prof R. Brown"
We are starting to follow up the work of John Shrimpton' Bangor thesis,
whose aim was the application of categorical methods in group theory, and
was partially published as "Some groups related to the symmetry of a
directed graph" JPAA 72 (1991) 303-318, and would be grateful for
information on publications in this area. We know of the paper by Bumby and
Latch, Internat J Math Sci 9 (1986) 1-16, P Ribenboim Algebraic structures
on graphs, Alg. Univ. 16 (1983) 105-123; Bill Lawvere, Qualitative
distictions between some toposes of generalized graphs, Cont. Math. 92,
261-299, (1989)
and would be grateful for further information. The aim of the project is to
use categorical methods to give insight into combinatorial problems,
constructions and questions.
A further aim was to understand Grothendieck's notion of "Teichmuller
groupoid" referred to in "Pursuing stacks", in the sense that if what he
asserted could be done was actually written up for surfaces, one should also
be able to do it for graphs, (or vice versa), and so obtain presentations
of a "symmetry groupoid" of a graph from some decomposition into smaller
pieces. I have no idea how to do this. Grothendieck's observation in this
area is quoted in my survey "From groups to groupoids" Bull London Math Soc.
19 (1987) 113-134. Part of this programme is "what should be the symmetry
object of a given structure?". New answers should be of general interest,
for obvious reasons.
Shrimpton's work showed that these questions lead to that of, for example,
inner automorphism of a directed graph, and the interpretation of this in
graph theoretic terms. There should be more in this area.
Ronnie Brown
Prof R. Brown
School of Mathematics,
University of Wales, Bangor
Gwynedd LL57 1UT
UK
Date: Fri, 4 Feb 1994 23:32:00 +0400 (GMT+4:00)
Subject: Union Conference
Date: 1 Feb 94 16:42:00 EST
From: "NIEFIELD, SUSAN "
NINTH BIENNIAL UNION COLLEGE MATHEMATICS CONFERENCE
Saturday and Sunday
April 9-10, 1994
Union College
Schenectady, New York
PRINCIPAL SPEAKERS
Zoltan Balogh
Andre Joyal
J. Peter May
The Mathematics Department of Union College is pleased to announce
its Ninth Biennial Mathematics Conference. In addition to the
lectures by the principal speakers, there will be concurrent sessions
in Algebraic Topology, Category Theory, and Set-Theoretic Topology.
SCHEDULE A detailed schedule will be mailed in March.
Friday Evening
Reception 9:00 - 11:00 PM (Bailey Hall, 2nd Floor)
Saturday
9:30 Registration, coffee, doughnuts
10:00 Parallel Sessions
11:30 Invited Address I: Andre Joyal
12:30 Lunch
2:00 Parallel Sessions
4:00 Coffee Break
4:30 Invited Address II: Peter May
6:00 Banquet and Party
Sunday
9:00 Coffee and doughnuts
9:30 Invited Address III: Zoltan Balogh
10:30 Parallel Sessions
12:00 Lunch
1:00 Parallel Sessions
3:30 Conference ends
REGISTRATION
Please pre-register for the conference before April 1, by contacting
one of the organizers listed below. There will be a $25 registration
fee ($15 for students) payable by mail or when you register at the
conference. Checks should be made out to "Union College Mathematics
Conference".
CONTRIBUTED TALKS
If you wish to give a talk in one of the sessions, please contact one
of the organizers of that session as soon as possible. Such talks should
be 25 minutes long.
ORGANIZERS
Algebraic Topology:
Brenda Johnson (johnsonb@unvax.union.edu)
Gaunce Lewis (gaunce@ichthus.syr.edu)
Category Theory:
Susan Niefield (niefiels@gar.union.edu)
Kimmo Rosenthal (rosenthk@gar.union.edu)
Set-Theoretic Topology:
Tim LaBerge (laberget@gar.union.edu)
ACCOMODATIONS
Blocks of rooms have been reserved at the Days Inn and the Holiday
Inn in Schenectady. To obtain one of these rooms at the rates listed
below, mention the Union College Mathematics Department when you call.
Both are within easy walking distance of the campus. The Ramada Inn
is also within walking distance of campus.
1. Holiday Inn (518) 393-4141
Single $55 Double $55
2. Days Inn (518) 370-0851
Single $45 Double $50
3. Ramada Inn (518) 370-7151
Single $63 Double $73
Department of Mathematics
Union College
Schenectady, NY 12308
(518) 388-6246
Date: Sun, 6 Feb 1994 23:07:21 +0400 (GMT+4:00)
Subject: 54th PSSL: preliminary announcement (latex)
Date: Fri, 4 Feb 1994 12:00:31 +0000 (GMT)
From: Edmund Robinson
\documentstyle[12pt,a4]{article}
\pagestyle{empty}
\begin{document}
\begin{center}
{\Large\bf Preliminary Announcement 54th PSSL}
\end{center}
The 54th meeting of the {\em Peripatetic Seminar on Sheaves
and Logic} will be held over the weekend of the 26th--27th March 1994
at the University of Sussex. As usual, we welcome talks on topics in
logic, category theory and related areas. Please publicise this
meeting amongst your colleagues.
\medskip
Further information about the exact venue, arrangements for meeting,
etc{.} will be distributed later. Information about travel, local
accommodation, participants and talks offered will be placed in the
Sussex anonymous ftp archive ({\tt ftp.cogs.sussex.ac.uk},
subdirectory {\tt pub/pssl}) as it becomes available. There is also a
mailbox {\tt pssl@cogs.sussex.ac.uk}, specifically for conference
mail. Use of this will considerably simplify our administration.
\medskip
We hope to arrange some sponsorship for the meeting, but participants
should note that we may be forced to charge a small registration fee
to cover the provision of food at the meeting.
\medskip
Return, preferably by email to {\tt pssl@cogs.sussex.ac.uk}, or to
\begin{tabbing}
aaaaaa\=aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa\=\kill
\> Edmund Robinson, \\
\> School of Cognitive and Computing Sciences,\\
\> University of Sussex,\\
\> Brighton, BN1 9QH\\
\> ENGLAND\\[1ex]
\> Tel: +44 273 678593\\
\> Fax: +44 273 671320
\end{tabbing}
We look forward to seeing you in Sussex. The local organizers, \\
Carolyn Brown,\\
Chris Mulvey,\\
Edmund Robinson,\\
Gavin Wraith
\medskip
\noindent\hrulefill
\medskip
I would like to attend the 54th PSSL:
\medskip
\noindent Name \dotfill
\noindent Return address \dotfill\\
\mbox{}\dotfill
\noindent Email \dotfill
\medskip
\begin{itemize}
\item[$\Box$] I wish to give a talk entitled
\noindent\mbox{}\dotfill
\noindent of 20 / 30 / 45 minutes. (Please send an abstract if possible).
\end{itemize}
\end{document}
Date: Sun, 6 Feb 1994 23:14:51 +0400 (GMT+4:00)
Subject: 54th PSSL: preliminary announcement (text)
Date: Fri, 4 Feb 1994 12:02:03 +0000 (GMT)
From: Edmund Robinson
Preliminary Announcement 54th PSSL
==================================
The 54th meeting of the Peripatetic Seminar on Sheaves and Logic will
be held over the weekend of the 26th--27th March 1994 at the
University of Sussex. As usual, we welcome talks on topics in logic,
category theory and related areas. Please publicise this meeting
amongst your colleagues.
Further information about the exact venue, arrangements for meeting,
etc. will be distributed later. Information about travel, local
accommodation, participants and talks offered will be placed in the
Sussex anonymous ftp archive (ftp.cogs.sussex.ac.uk, subdirectory
pub/pssl) as it becomes available. There is also a mailbox
"pssl@cogs.sussex.ac.uk", specifically for conference mail. Use of this
will considerably simplify our administration.
We hope to arrange some sponsorship for the meeting, but participants
should note that we may be forced to charge a small registration fee
to cover the provision of food at the meeting.
Return, preferably by email to "pssl@cogs.sussex.ac.uk", or to
Edmund Robinson,
School of Cognitive and Computing Sciences,
University of Sussex,
Brighton, BN1 9QH
ENGLAND
Tel: +44 273 678593
Fax: +44 273 671320
We look forward to seeing you in Sussex. The local organizers,
Carolyn Brown,
Chris Mulvey,
Edmund Robinson,
Gavin Wraith
--------------------------------------------------------------------------
I would like to attend the 54th PSSL:
Name:
Return address:
Email:
[] I wish to give a talk entitled:
of 20 / 30 / 45 minutes. (Please send an abstract if possible).
Date: Sat, 12 Feb 1994 16:39:13 +0400 (GMT+4:00)
Subject: Re: cantor-bernstein
Date: Sat, 12 Feb 94 16:35:35 GMT
From: Dusko Pavlovic
Hi. I was away for a while and read the postings from the last couple
of months only now. Hence this delayed reaction on Peter Freyd's
comments from January on the Cantor-Schroeder-Bernstein theorem.
> I trust that CSB holds for any boolean topos. (Anyone want to confirm?)
The other way around, the validity of CSB in a topos ALMOST implies
that the topos is boolean. "Almost" means a slightly stronger version
needs to be considered:
(CSB*) if f:A->B and g:B->A are monics
then there is an iso h:A->B
such that for every x in A holds
h(x) = f(x) or h(x) = g^{-1}(x)
The usual set-theoretical proof of CSB (which goes through in boolean
toposes) actually yields (CSB*). In return, (CSB*) implies
booleanness. We assume that every topos has an object of natural
numbers.
PROPOSITION. A topos is boolean if and only if it satisfies (CSB*).
PROOF of the "if" part: Let X be a subobject of Y. We use (CSB*) to
construct its complement. Define
A = YxN
B = X+(YxN)
(where N is the object of natural numbers). Let f:A-->B be the
inclusion, and g:B-->A the unique arrow which maps X into the first
copy of Y in A, while it takes the copies of Y contained in B into the
second one, the third one and so on. (Formally, g is defined using the
universal property of coproducts, and the structure of N.) It is clear
that both f and g are monics.
By (CSB*), there is a bijection h:A-->B with h(x)=f(x) or
h(x)=g^{-1}(x). This means that the union of
C = {z | h(z)=f(z)} and
D = {z | h(z)=g^{-1}(z)}
covers A. On the other hand, the definitions of f and g imply that C
and D are disjoint. So they are complements in A. By pulling back C and D
along the inclusion of Y iso Yx{0} in A, we get two complementary
subobjects of Y. The claim is now that the intersection (pullback) X'
of D and Yx{0} is isomorphic to X -- so that the intersection of C and
Yx{0} yields the complement of X in Y.
First of all, each element of D must surely be in the image of
g. Thus, if is in D, then y must be in X. So X' is contained in
Xx{0}. The other way around, since h(C) is contained in f(C), the
intersection in B of X and h(C) must be empty. Therefore, X must be
contained in h(D): indeed, h is an iso, and the direct images h(C) and
h(D) must be complementary in B. But h(D) is contained in
g^{-1}(D). So X is in g^{-1}(D), i.e. g(X) is in D. Hence, Xx{0}=g(X)
is contained in X'.
Regards to all,
Dusko
--
Date: Sun, 13 Feb 1994 13:46:02 +0400 (GMT+4:00)
Subject: Re: cantor-bernstein
Date: Sun, 13 Feb 94 09:11:14 EST
From: Peter Freyd
Nice theorem from Dusko Pavlovic. But there's a better one:
If a topos with NNO is CSB then it's boolean (where CSB means:
if two objects be retracts of each other they are
necessarily isomorphic).
Let _A_ be a topos with NNO, N. Let D be its object that
"classifies dense subobjects", that is, the object with a
distinguished point t:1 -> D such that for every double-negation
dense subobject A' -> A there is a unique map A -> D such that
A' -> A is the inverse image of t:1 -> D. (We may construct D as
the double-negation closure of the t in Omega.)
It suffices to show that D is isomorphic to 1.
CSB implies that 1 + N x D and N x D are isomorphic. Let
1 + N x D -> N x D be an isomorphism. Preceding by the injection
of 1 and following by the projection onto N we obtain
1 -> 1 + N x D -> N x D -> N which yields a countable
partitioning of 1, that is, an (internally definable) sequence of
pairwise-disjoint subobjects whose union is 1. Let U be an
arbitrary one of these subobjects. It suffices to show that the slice
topos, *A*/U, is boolean. Without loss of generality, then, we may
assume that U = 1, that is, we may assume that the map
1 -> 1 + N x D -> N x D -> N is itself the inclusion map of a
complemented subobject of N. Restated: the isomorphism carries the
complemented copy of 1 in 1 + N x D to a complemented copy of D
in N x D forcing D to have a complemented copy of 1. It is left
as an exercise to see that the only way that D can have a
complemented copy of 1 is if it is that copy of 1.
Can we remove the need for NNO? Nope. Let M be any finite monoid.
The category of finite M-sets is a topos. Every endomorphism monoid
is finite, hence every monic (or epic) endo is an auto and CSB more
than holds.
Date: Sun, 13 Feb 1994 17:06:02 +0400 (GMT+4:00)
Subject: Re: cantor-bernstein
Status: O
Date: Sun, 13 Feb 94 15:48:26 EST
From: Peter Freyd
Alright. The exercise goes as follows: suppose that s:1 -> D names a
complemented subobject. Let V be the complemented subterminator
obtained by intersecting s and t and let W be its complement.
In *A*/W we have that s and t name disjoint subobjects but t
names a dense subobject of D hence W = 0. Because V = 1 we have
s = t naming an object both dense and complemented hence entire.
Date: Fri, 18 Feb 1994 14:21:27 +0400 (GMT+4:00)
Subject: new book
Date: Fri, 18 Feb 1994 12:35:17 +0100 (MET)
From: Jiri Rosicky
Preliminary announcement
The following book will be available shortly:
Locally Presentable and Accessible Categories
J.Adamek, J.Rosicky
Cambridge University Press
London Mathematical Society Lecture Notes Series 189
ISBN 0 521 42261 2
330 pp.
Abstract: The concepts of a locally presentable category and an accessible
category have turned out to be useful in formulating connections between
universal algebra, model theory, logic and computer science. The aim of this
book is to provide an exposition of both the theory and the applications
of these categories at a level accessible to graduate students.
Firstly the properties of lambda-presentable objects, locally
lambda-presentable categories, and lambda-accessible categories are
discussed in detail, and the equivalence of accessible and sketchable
categories is proved. The authors go on to study categories of algebras
and prove that Freyd's essentially algebraic categories are precisely
the locally presentable categories. In the final chapters they treat some
topics in model theory and some set theoretical aspects.
Contents: 0. Preliminaries, 1. Locally presentable categories, 2. Accessible
categories, 3. Algebraic categories, 4. Injectivity classes, 5. Categories
of models, 6. Vopenka's principle, Appendix: Large cardinals, Open problems.