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From rrosebru@mta.ca Thu Sep  1 09:30:55 2005 -0300
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Date: Thu, 1 Sep 2005 07:53:54 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
Message-Id: <200509011153.j81BrsJ4002446@saul.cis.upenn.edu>
To: categories@mta.ca, tl@maths.gla.ac.uk
Subject: categories: Re: Preprint: A simple description of Thompson's group F
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  There's a good chance that the characterization of Thompson's group F
  (not to mention its name) was set forth in the paper reviewed below
  (the authors of which became aware of R.J.Thompson's priority via
  this review).


Freyd, Peter; Heller, Alex
Splitting homotopy idempotents. II.
J. Pure Appl. Algebra 89 (1993), no. 1-2, 93--106.

A preliminary version of this paper was in the reviewer's hands in
1979 and was then of uncertain age. The authors have done a service in
publishing it (in somewhat revised form) belatedly.

The object of study is a free homotopy idempotent $f \colon X \to X$;
this means that $f$ is freely (base point not necessarily preserved
during the homotopy) homotopic to $f^2 \equiv f \circ f$. This $f$ is
said to split if there are maps $d \colon X \to Y$ and $u \colon Y \to
X$ such that $d \circ u \simeq \text{id}_Y$ and $u \circ d \simeq f$,
where $\simeq$ denotes free homotopy.

They construct a group $F$ and an endomorphism $\phi \colon F \to F$
such that, for a certain $\alpha_0 \in F$, $\phi^2(7) =
\alpha^{-1}_0\phi(7)\alpha_0$. The induced map $g \colon K(F,1) \to
K(F,1)$ is a homotopy idempotent which does not split; and it is
universal in the sense that it maps "canonically" into any homotopy
idempotent, and the corresponding homomorphism $F \to \pi_1(X)$ is
monic if and only if $f$ does not split.

This group $F$ is shown to be finitely presentable, has simple
commutator subgroup, is a totally ordered group and contains a copy of
its own infinite wreath-product. Every abelian subgroup is free
abelian, and every subgroup is either finite-rank free abelian or
contains an infinite-rank free abelian subgroup.

\{Reviewer's remarks: (1) While the authors acknowledge that some of
the above is due independently to J. Dydak
\ref[Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25
(1977), no. 1, 55--62; MR0442918 (56 \#1293)], they fail to mention
that priority for this group is generally given to R. J. Thompson
\ref[R. J. Thompson and R. McKenzie, in Word problems (Irvine, CA,
1969), 457--478, North-Holland, Amsterdam, 1973; MR0396769 (53
\#629)], who introduced $F$ and seemed to know many of its properties
in the late 1960s. Closely related to $F$ are Thompson's finitely
presented infinite simple groups. (2) Subsequently, as acknowledged by
the authors, much more became known about this extraordinary group. To
help the reader know what we are discussing, we mention that $F$ is
often known as "the Richard Thompson group"; also as the "Freyd-Heller
group", the "Dydak-Minc group" and (incorrectly, but because of later
work on $F$) as the "Brown-Geoghegan group". (3) The origin of the
curious name "$F$" was explained to the reviewer by one of the authors
as standing for "free", as in "free homotopy idempotent".\}

Reviewed by Ross Geoghegan




From rrosebru@mta.ca Fri Sep  2 09:36:31 2005 -0300
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Date: Thu, 1 Sep 2005 10:56:18 -0700 (PDT)
From: Joseph Goguen <goguen@cs.ucsd.edu>
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Subject: categories: Preprint: Information integration in institutions
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The paper whose abstract is given below is available at

    http://www.cs.ucsd.edu/~goguen/pps/ifi04.pdf

and will appear in a memorial volume for Jon Barwise
sometime in 2006, edited by Larry Moss.


Information Integration in Institutions
   Joseph A Goguen
   Department of Computer Science  and Engineering
   University of California at San Diego, USA

This paper unifies and/or generalizes several approaches to information,
including the information flow of Barwise and Seligman, the formal conceptual
analysis of Wille, the lattice of theories of Sowa, the categorical general
systems theory of Goguen, and the cognitive semantic theories of Fauconnier,
Turner, Gardenfors, and others.  Its rigorous approach uses category theory
to achieve independence from any particular choice of representation, and
institutions to achieve independence from any particular choice of logic.
Corelations and colimits provide a general formalization of information
integration, and Grothendieck constructions extend this to several kinds of
heterogeneity.  Applications include modular programming, Curry-Howard
isomorphism, database semantics, ontology alignment, cognitive semantics, and
more.




From rrosebru@mta.ca Fri Sep  2 09:36:32 2005 -0300
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Date: Thu, 1 Sep 2005 12:17:45 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
Message-Id: <200509011617.j81GHj1q020427@saul.cis.upenn.edu>
To: categories@mta.ca, tl@maths.gla.ac.uk
Subject: categories: Re: Preprint: A simple description of Thompson's group F
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Marcelo and Tom write

  We show that Thompson's group F is the symmetry group of the "generic
  idempotent".  That is, take the monoidal category freely generated by an
  object A and an isomorphism A \otimes A --> A; then F is the group of
  automorphisms of A.

Tom has pointed out to me that the review of the old Freyd/Heller I
posted give no hint of its relevance. Therefor this:

  F was defined (40 years ago) as the initial model for a group with
  an endomorphism that's conjugate to its square.

More formally: consider the equational theory that adds to the theory
of groups a constant, s, and a unary operator  e, subject to two
further equations:

    e(xy) = (ex)(ey)             "e is a endomorphism"
    s(ex) = (e(ex))s             "e is a conjugacy-idempotent"

The initial algebra for this theory is the group  F.

(If one insists on removing the type-error in the last sentence, then
try "the initial algebra for this theory when subjected to the
forgetful functor back to groups is  F.")

If one defines a sequence of elements   s_n = e^n(s)  they clearly
generate  F (as a group) and it isn't hard to see that a complete set
of relations for  F (as a group) is the doubly-infinite family

     s_a s_b = s_{b+1} s_a

one such equation for each  a < b. (It took me ten years to find a
proof that just two of these equations imply all the others.)



From rrosebru@mta.ca Mon Sep  5 17:14:04 2005 -0300
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Date: Mon, 05 Sep 2005 20:44:10 +0100
From: cie06@swansea.ac.uk
To: categories@mta.ca
Subject: categories: CiE 2006, Call for Papers
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                               CiE 2006
                     Computability in Europe 2006 :
              Logical Approaches to Computational Barriers
                        30 June - 5 July 2006
                         Swansea University
                   http://www.cs.swansea.ac.uk/cie06/

                          CALL  FOR  PAPERS

                      Deadline: DECEMBER 15, 2005

CiE 2006 is the second of a new conference series on Computability
Theory and related topics which started in Amsterdam in 2005.  CiE 2006
will focus on (but not be limited to) logical approaches to
computational barriers:
- practical and feasible barriers, e.g., centred around the P vs. NP
   problem;
- computable barriers connected to models of computers and
   programming languages;
- hypercomputable barriers related to physical systems.

Tutorials will be given by:
   Samuel R. Buss (San Diego)
   Julia Kempe (Paris)

Invited Speakers include:
   Jan Bergstra (Amsterdam)
   Luca Cardelli (Microsoft Cambridge)
   Jan Krajicek (Prague)
   Elvira Mayordomo Camara (Zaragoza)
   Istvan Nemeti (Budapest)
   Helmut Schwichtenberg (Munich)
   Andreas Weiermann (Utrecht)

The Programme Committee cordially invites all researchers (European
and non-European) in the area of Computability Theory to submit their
papers (in PDF-format, at most 10 pages) for presentation at CiE 2006.
We particularly invite papers that build bridges between different
parts of the research community.  Since women are underrepresented in
mathematics and computer science, we emphatically encourage
submissions by female authors.

The proceedings are intended to be published within Springer's LNCS
series. Important dates are:

Submission Deadline:         December 15th, 2005.
Notification of Authors:     February 15th, 2006.
Deadline for Final Version:     March 15th, 2006.

Programme Committee:
   Samson Abramsky  (Oxford)
   Klaus Ambos-Spies (Heidelberg)
   Arnold Beckmann (Swansea, co-chair)
   Ulrich Berger (Swansea)
   Olivier Bournez (Nancy)
   Barry Cooper (Leeds)
   Laura Crosilla (Firenze)
   Costas Dimitracopoulos (Athens)
   Abbas Edalat (London)
   Fernando Ferreira (Lisbon)
   Ricard Gavalda (Barcelona)
   Giuseppe Longo (Paris)
   Benedikt Loewe (Amsterdam)
   Yuri Matiyasevich (St.Petersburg)
   Dag Normann (Oslo)
   Giovanni Sambin (Padova)
   Uwe Schoening (Ulm)
   Andrea Sorbi (Siena)
   Ivan Soskov (Sofia)
   Leen Torenvliet (Amsterdam)
   John Tucker (Swansea, co-chair)
   Peter van Emde Boas (Amsterdam)
   Klaus Weihrauch (Hagen)

Confirmed sponsors:
   British Logic Colloquium (BLC)
   Kurt Goedel Society (KGS)
   Welsh Development Agency (WDA)

For more information about the conference please check the CiE
conference series http://www.illc.uva.nl/CiE/ and our web page
http://www.cs.swansea.ac.uk/cie06/.





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Date: Mon, 05 Sep 2005 15:53:32 +0100
From: Philip Wadler <wadler@inf.ed.ac.uk>
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To: categories@mta.ca
Subject: categories: Eighth International Symposium on Functional and Logic Programming
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[FLOPS benefits from an eclectic mix of FP and LP papers, one of the few
venues where the two communities get together.   It should be a
congenial meeting, situated under Mt Fuji.  Do come!   -- P]

                            First Call For Papers

   Eighth International Symposium on Functional and Logic Programming
                                FLOPS 2006

                               April 24--26
                            Fuji Susono, JAPAN

                      Submission deadline: November 11
                  http://hagi.is.s.u-tokyo.ac.jp/FLOPS2006

FLOPS is a forum for research on all issues concerning declarative
programming, including functional programming and logic programming,
and aims to promote cross-fertilization between the two paradigms.
Previous FLOPS meetings were held in Fuji Susono (1995), Shonan
Village (1996), Kyoto (1998), Tsukuba (1999), Tokyo (2001), Aizu
(2002), and Nara (2004).

TOPICS

FLOPS solicits original papers in all areas of functional and logic
programming, including (but not limited to):

     Declarative Pearls: new and excellent declarative programs with
     illustrative applications;

     Language issues: language design and constructs, programming
     methodology, integration of paradigms, interfacing with other
     languages, type systems, constraints, concurrency and distributed
     computing;

     Foundations: logic and semantics, rewrite systems and narrowing,
     type theory, proof systems;

     Implementation issues: compilation techniques, memory management,
     program analysis and transformation, partial evaluation,
     parallelism;

     Applications: case studies, real-world applications, graphical user
     interfaces, internet applications, XML, databases, formal methods,
     and model checking.

For 2006, we wish to particularly encourage papers on new application
areas, including security, bioinformatics, and quantum computation.

The proceedings will be published as a LNCS volume.  The proceedings
of the previous meeting (FLOPS2004) were published as LNCS2998.

SUBMISSION

Submissions must be unpublished and not submitted for publication
elsewhere. Work that already appeared in unpublished or informally
published workshops proceedings may be submitted. Submissions should
fall into one of the following categories:

Regular research papers: they should describe new results and will be
judged on originality, correctness and significance.

System descriptions: they should contain a link to a working system
and will be judged on originality, usefulness and design.

All submissions must be written in English and can be up to 15
proceedings pages long. Authors are strongly encouraged to use LaTeX2e
and the Springer llncs class file, available at
http://www.springer.de/comp/lncs/authors.html

Regular research papers should be supported by proofs and/or
experimental results.  In case of lack of space, this supporting
information should be made accessible otherwise (e.g. a link to a web
page, or an appendix).

Submission is Web-based and under preparation.  Please visit
http://hagi.is.s.u-tokyo.ac.jp/FLOPS2006/

INVITED SPEAKERS

     Peter Van Roy (Louvain, Belgium)
     other speakers to be decided

CO-CHAIRS

     Philip Wadler (Edinburgh, UK)
     Masami Hagiya (Tokyo, Japan)

PC MEMBERS

     Peter Selinger (Dalhousie, Canada)
     Manuel Hermenegildo (New Mexico & Madrid, US & Spain)
     Eijiro Sumii (Tohoku, Japan)
     Konstantinos Sagonas (Uppsala, Sweden)
     Jacques Garrigue (Nagoya, Japan)
     Peter Thiemann (Freiburg, Germany)
     David Warren (Stony Brook, US)
     Gabrielle Keller (UNSW, Sydney, Australia)
     Alain Frisch (INRIA Roquencourt, France)
     Veronica Dahl (Simon Fraser, Canada)
     Ken Satoh (NII, Tokyo, Japan)
     Naoyuki Tamura (Kobe, Japan)
     Michael Rusinowitch (INRIA Lorraine, France)
     Vincent Danos (Paris, France)

IMPORTANT DATES

    Submission deadline: November 11
    Author notification: January 6
    Camera-ready copy: January 20

PLACE

The meeting will be held at Fuji Institute of Education and Training
(http://www.fujiken.gr.jp/) located in Fuji Susono, JAPAN, where the
first FLOPS was held.  It is famous of its view to Mt. Fuji.

Previous FLOPS:

     FLOPS 2004, Nara: http://logic.is.tsukuba.ac.jp/FLOPS2004/
     FLOPS 2002, Aizu: http://www.ipl.t.u-tokyo.ac.jp/FLOPS2002/
     FLOPS 2001, Tokyo: http://www.ueda.info.waseda.ac.jp/flops2001/

SPONSOR

     University of Tokyo

IN COOPERATION (pending)

     ACM SIGPLAN
     Japan Society for Software Science and Technology (JSSST)
     Association for Logic Programming (ALP)
     Asian Association for Foundation of Software (AAFS)

INQUIRIES to

     Masami Hagiya
     hagiya@is.s.u-tokyo.ac.jp



From rrosebru@mta.ca Wed Sep  7 07:19:50 2005 -0300
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Date: Tue, 6 Sep 2005 16:21:36 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
Message-Id: <200509062021.j86KLanB027117@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: categories: Memorial service for John
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On August 10 I received the following information about John Isbell's
death.

  A memorial service is now being planned and will most likely occur
  at Forest Lawn on Saturday, September 10. No other details are
  available now; an obituary should appear in the Buffalo News today
  or later this week.

No obit has appeared (paid or ortherwise) and I have not succeeded in
finding any further information about the service.

Does anyone know?

   Peter



From rrosebru@mta.ca Wed Sep  7 19:56:19 2005 -0300
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Date: Wed, 7 Sep 2005 18:46:01 -0400 (EDT)
From: Robert Seely <rags@math.mcgill.ca>
To: Categories List <categories@mta.ca>
Subject: categories: News from Tulane?
Message-ID: <Pine.LNX.4.44.0509071842470.10579-100000@prism.math.mcgill.ca>
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Just wondering if anybody here has had any news from our collegues from
Tulane (eg Mike Mislove)?  Are they safe, did they evacuate before the
hurricane, what's in store for them now?  (I know the university is shut -
and I guess probably for the forseeable future, though that's also a
question.)  Obviously one wishes them all the best in this impossibly
difficult time.

-= rags -=

-- 
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>




From rrosebru@mta.ca Thu Sep  8 16:34:07 2005 -0300
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From: Colin McLarty <colin.mclarty@case.edu>
To: categories@mta.ca
Message-ID: <503ca6503bb9.503bb9503ca6@cwru.edu>
Date: Thu, 08 Sep 2005 07:52:34 -0400
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Subject: categories: Noether and fast thinking
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Somewhere MacLane published a part of a letter he sent his mother from
Goettingen where he said that Fraulein Noether "thought fast and spoke
faster" or something like that.  I have looked at every mention of
Noether by him that I can think of without re-locating this one.  Does
anyone know where it is?

thanks, Colin



From rrosebru@mta.ca Thu Sep  8 16:34:07 2005 -0300
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Date: Thu, 08 Sep 2005 00:44:47 -0400
From: Fred E.J.Linton <fejlinton@usa.net>
To: <categories@mta.ca>
Subject: categories: Re: Memorial service for John
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Hello, Peter, and all Categories readers,

Googling "Buffalo News obituary John Isbell" today brought this up:

> John R. Isbell Ph.D.    =

> =

> August 6, 2005 age 74. Dear father of Margaret M. Thornborough of =
> Bristol, England, John C. Isbell of Bloomington, IN, and Brecht W. =
> Isbell of Los Angeles, CA.; grandfather of Zelie Thornborough and =
> Alexander L. L. Thornborough both of England; brother of Frances W. =
> Isbell of Weslaco, TX and the late Robert O. Isbell. There will be no =
> prior visitation. A Memorial Service will be held Saturday, September =
> 10, 2005 at 4 PM at the Chapel of Forest Lawn Cemetery. Friends =
> invited. Mr. Isbell was a Math Professor at SUNY @ Buffalo from 1969-
> 1999. Arrangements by AMIGONE FUNERAL HOME INC. Online guest register =
> at www.Amigone.com =

>
> Published in the Buffalo News on 8/28/2005.  =


[URL (all one line; best turn off Active-X):

http://legacy.com/BuffaloNews/LegacySubPage2.asp?Page=3DLifeStory&PersonI=d=3D14948085

]


------ Original Message ------
Received: Wed, 07 Sep 2005 06:25:08 AM EDT
From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: categories: Memorial service for John

> On August 10 I received the following information about John Isbell's
> death.
> =

>   A memorial service is now being planned and will most likely occur
>   at Forest Lawn on Saturday, September 10. No other details are
>   available now; an obituary should appear in the Buffalo News today
>   or later this week.
> =

> No obit has appeared (paid or ortherwise) and I have not succeeded in
> finding any further information about the service.
> =

> Does anyone know?
> =

>    Peter
> =

> =

> =







From rrosebru@mta.ca Thu Sep  8 16:34:07 2005 -0300
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Date: Wed, 07 Sep 2005 16:13:17 -0700
From: Todd Wilson <twilson@csufresno.edu>
Subject: categories: Follow-ups to [HP89]?
To: categories@mta.ca
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In the introduction to their 1989 article,

    JME Hyland & AM Pitts, "The Theory of Constructions: Categorical
    Semantics and Topos-Theoretic Models", Contemp. Math. 92 (1989), 137
    - 199,

the authors say, "Clearly this paper is only a beginning."  Can someone
recommend follow-ups to this paper published in the intervening 15 years?

Todd Wilson
Department of Computer Science
California State University, Fresno



From rrosebru@mta.ca Thu Sep  8 16:38:17 2005 -0300
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To: Categories List <categories@mta.ca>
Subject: categories: Re: News from Tulane?
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[Note from moderator: Thanks to the many who replied, several answers are
combined below.]

Robert Seely wrote:

>Just wondering if anybody here has had any news from our collegues from
>Tulane (eg Mike Mislove)?  Are they safe, did they evacuate before the
>hurricane, what's in store for them now?  (I know the university is shut -
>and I guess probably for the forseeable future, though that's also a
>question.)  Obviously one wishes them all the best in this impossibly
>difficult time.
>
>-= rags -=
>

Some Tulane students have been accepted here at Penn
and we have picked up 2 post-docs who would've been there.

jim


Date: Wed, 7 Sep 2005 16:57:46 -0700 (PDT)
From: Dominic Hughes <dominic@theory.stanford.edu>
Subject: Re: categories: News from Tulane?

I know second-hand that Mike Mislove and James (aka Ben) Worrell got away
safely.

Dominic

From: Michael Fourman <Michael.Fourman@ed.ac.uk>
Subject: Re: News from Tulane?
Date: Thu, 8 Sep 2005 06:06:16 +0100

I checked this a few days ago. Google search for
Mislove Tulane Katrina
turns up the following

http://atgdev.elsevier.com/blogs/cleonard/?p=3D34

Katrina and Tulane
September 2nd, 2005
I=92ve just had an email from Mike Mislove, editor of Electronic Notes
=20=

in Theoretical Computer Science. He is usually based in New Orleans =20
at Tulane University, but is currently safe in Memphis having =20
evacuated before the hurricane hit land. Apparently it will be some =20
time before Tulane will be back to normal - daily updates on the =20
situation can be found on a special website.

As a result of the recent devastation, the usual Tulane mail servers =20
are not working. If you wish to contact Mike, please use his gmail =20
account.



From: Don Sannella <dts@inf.ed.ac.uk>
Date: Thu, 8 Sep 2005 08:57:19 +0100
Subject: Re: News from Tulane?


I recently heard from Chris Leonard at Elsevier -- where Mike Mislove
is editor-in-chief of Electronic Notes in Theoretical Computer Science
-- that Mike got out before the hurricane hit, is safe in Memphis, and
can be reached at michael.mislove@gmail.com.  I don't know anything
about anybody else at Tulane.

Don Sannella

From: Martin Escardo <m.escardo@cs.bham.ac.uk>
Date: Thu, 8 Sep 2005 09:39:29 +0100
Subject: Re: News from Tulane?

I know that Mike Mislove and Karl Hofmann safely rellocated themselves
before the hurricane. Because Mike didn't mention, I assume that all
other colleagues are safe, but I don't really know.


Date: Thu, 08 Sep 2005 02:46:21 -0700
From: Dusko Pavlovic <dusko@kestrel.edu>
Subject: Re: categories: News from Tulane?

mike mislove is in a hotel in memphis, tn. they evacuated before the
hurricane, but times are difficult. we hope that he and his family will
visit palo alto as soon as the circumstances allow.

ben worrell was in canada, scheduled to return to new orleans on the
morning of katrina. he was to start at oxford soon, so i hope that he is
now in UK.

i have no idea about the others.

-- dusko





From rrosebru@mta.ca Mon Sep 12 17:01:04 2005 -0300
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Subject: categories: John Isbell Memorial
Date: Mon, 12 Sep 2005 15:37:56 -0400
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Hi,

Please post the following note from the UB Mathematics Department (not
me).

Since postings on the catnet have sought further information on the memorial for
John Isbell, the Mathematics Department of the University at Buffalo, SUNY would
like to assure the community that the service and announcements were handled
according to the wishes of the family.  The service itself was held on Saturday
afternoon and remarks about John's remarkable intellectual breadth and extensive
mathematical accomplishments were made by Professors J. Duskin, F.W. Lawvere, S.
Schanuel, and S. Williams.  The family is preparing an obituary, with more
detail than the death notice copied to the catnet by F. Linton, and we will
forward it to the catnet when it appears.




From rrosebru@mta.ca Fri Sep 16 11:48:29 2005 -0300
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Date: Fri, 16 Sep 2005 13:19:10 +0100
To: categories@mta.ca
Subject: categories: stable flatness and exponentiation
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I have three questions:

Question 1: Is there an intrinsic characterization of the stably flat
maps of locales, i.e., those continuous maps f: X->Y such that

   id_Z x f: Z x X -> Z x Y

is flat for all Z?

(Recall that f: X->Y is flat iff the right adjoint f_*: OX -> OY of
f^*: OY -> OX preserves finite joins. Preservation of merely the empty
join amounts to density, and hence flatness is a rather strong form of
density. For example, the embedding of a completely regular locale
into its Stone-Cech compactification is strongly dense in this sense.)

I don't think all flat maps are stably flat, but I may be
mistaken. Specifically, for a locale X let JX denote the spectral
locale defined by

   OJX = ideals of OX.

Then there is a sublocale embedding eta: X->JX defined by

   eta^*(I) = join I,

which is known to be flat.

Question 2: Is eta: X->JX stably flat for every X?

If so, then compact Hausdorff locales would be exponentiating. Here a
locale Y is called exponentiating iff the exponential Y^X exists for
every X. Recall also that X is exponentiable iff the exponential Y^X
exists for every Y, and that this is the case iff the frame OX is a
continuous lattice.

Now, JX is exponentiable because OJX is an algebraic lattice. We claim
that if eta_X were stably flat, then for every compact Hausdorff
locale Y, the exponential Y^JX would have the universal property of an
exponential Y^X with respect to a suitably defined evaluation map and
a suitable construction of exponential transposition.

Define e: Y^JX x X -> Y as the restriction of the original evaluation
map ev: Y^JX x JX -> Y, that is

             id x eta                ev
   Y^JX x X ----------> Y^JX -> JX -------> Y.
            ------------------------------>
                          e

Now, to show that the pair (Y^JX,e) has the universal property of an
exponential Y^X, given f: Z x X -> Y, we have to construct a transpose
f': Z -> Y^JX such that e(f' x id_X) = f.

Consider the diagram


              id_Z x eta_X
       Z x X --------------> Z x JX
          \                   .
           \                 .
            \               .
             \             .
              \           .
            f  \         .  f''
                \       .
                 \     .
                  \   .
                   v v
                    Y.

If eta_X were stably flat, then, by "Joyal's Lemma", which says that
compact Hausdorff locales are orthogonal to flat embeddings, there
would be a unique f'' making the diagram commute.

Then its Y^JX-transpose f': Z -> Y^JX with respect to the original
evaluation map would give the unique required Y^X-transpose of our
given f: Z x X -> Y with respect to our constructed evaluation map, as
an easy calculation shows, using the universal property of Y^JX with
respect to the original evaluation map.

This shows that if Question 2 had a positive answer then compact
Hausdorff locales would be exponentiating.

Question 3. But surely compact Hausdorff locales cannot possibly be
exponentiating, can they?

(These questions make sense for topological spaces too. )

MHE





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Date: Sat, 17 Sep 2005 10:30:53 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
Message-Id: <200509171430.j8HEUraX013068@saul.cis.upenn.edu>
To: categories@mta.ca
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                     Copyright 2005 TSL Education Limited
                     The Times Higher Education Supplement

                               September 16, 2005

SECTION: LETTER; No.1709; Pg.17

LENGTH: 448 words

HEADLINE: Allow Beautiful Minds To Thrive

BYLINE: Ronnie Brown

BODY:

The British Association for the Advancement of Science warns that the research
assessment exercise does not recognise the importance of the public
communication of science ("Scientists want time to talk", September 9).
Experience in the mathematics faculty at Bangor was that the Teaching Quality
Assessment did not recognise it either.

This is a kind of travesty. Exploration, exposition and communication have for
centuries been recognised as essential to the progress of science.

Where would we be without Euclid's marvellous compilation of the geometry of his
day? Galileo, Faraday, Poincare, Klein, Hilbert, Einstein, Hoyle and Feynman
have all made public communication, and often disagreement with authority, an
important part of their work.

Our aim for the popularisation of mathematics has been, to modify Science
Minister Lord Sainsbury's words in the same issue of The Times Higher, to show
the public, students and the Government not only the important role that
mathematics plays in society, but also how it evolves.

Mathematics progresses partly through the solution of problems, but also through
clarification and good exposition, providing a developing language for
description, verification, deduction and calculation. It makes the difficult
easy. It works over a long timescale. It shows new possibilities through gradual
conceptual advance. It formulates new problems.

So mathematics is a foundation of the modern technological society. It is a
considerable challenge to try to show advanced mathematics from an elementary
viewpoint.

Some results of our work in popularisation of mathematics at Bangor over the
past 20 years may be seen on our website www.popmath.org.uk. We have had strong
support from, among others, the patrons of the sculptor John Robinson, for
promoting his Symbolic Sculptures.

An unplanned consequence has been sculptures by Robinson at, for example,
Bangor, Cambridge, Durham and Macquarie universities.

This supports the aim of associating mathematics and science with art, and
demonstrates art as a mode of symbolising an idea.

Work in communicating to children and the general public ideas in mathematics
has helped us to analyse and express our programme, to communicate mathematical
concepts to fellow scientists and students, and so to interdisciplinary
collaboration.

For the future of the UK, the public communication of science and mathematics
should be supported financially and in career structure, and be part of the
assessment of the success of a university and of the vitality of research and
teaching teams.

Ronnie Brown Emeritus professor of mathematics University of Wales, Bangor





From rrosebru@mta.ca Sun Sep 18 10:27:21 2005 -0300
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To: categories@mta.ca
Subject: categories: Re: stable flatness and exponentiation
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Martin Escardo writes:
 > Question 3. But surely compact Hausdorff locales cannot possibly be
 > exponentiating, can they?
 >
 > (These questions make sense for topological spaces too. )

Clarification:

Alex Simpson points out, in a reply not sent to the list, that in the
ambient category of topological spaces, compact Hausdorff spaces are
not exponentiating. For example the exponential 2^(N^N) doesn't exist,
where 2 is the two-point discrete space and N is the discrete space of
natural numbers, and, of course, the exponential N^N does exist. In
fact, Alex emphasized this years ago, when I subjected him to the
ideas presented in the previous message.

The point is that the locale product of two (sober) topological spaces
doesn't coincide with the topological product, and hence exponentials
are potentially different, as they are defined with respect to
products.

You may say: well, in any case, it is a fact that a sober space is
exponentiable in Top iff it is exponentiable in Loc, so Alex's
counter-example should work in Loc too.  But then I reply: this
coincidence has to do with the fact that the locale product coincides
with the topological product if one of the factors is locally
compact. In our case, because the exponent is NOT necessarily locally
compact, this coincidence fails. We are considering exponentiating
rather than exponentiable spaces. So, in principle, it may be that
compact Hausdorff locales are exponentiating in Loc - although I very
much doubt that this would be the case, as should be clear from the
previous message. The point is that I just don't know, and I am
looking forward to be enlightened after my failed attempts to decide
the question either way.

In light of Alex's observation, my conclusion to the previous message
should have been:

> Question 3. But surely compact Hausdorff locales cannot possibly be
> exponentiating, can they?

> It would be rather amazing if they were, because compact Hausdorff
> spaces are not exponentiating in the category of topological
> spaces. But the known arguments in the case of topological spaces
> don't seem to apply here.

One further comment: it is plausible that eta: X->JX is stably flat
for X exponentiable. If Alex's topological counter-example applies to
locales, then eta_X is not stably flat for X=N^N.

MHE




From rrosebru@mta.ca Mon Sep 19 13:28:58 2005 -0300
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To: categories@mta.ca
From: David Roberts <droberts@maths.adelaide.edu.au>
Subject: categories: Question re lax crossed modules
Date: Mon, 19 Sep 2005 12:44:30 +0930
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I have been looking at categorical groups a little and was wondering
what a lax crossed module is. A search through various databases has
turned up nothing. It would seem that they should be like crossed
modules but only satisfy a weakened equivariance property.

Any pointers toward a definition would be great.


------------------------------------------------------------------------
--
David Roberts
School of Mathematical Sciences
University of Adelaide SA 5005
------------------------------------------------------------------------
--
droberts@maths.adelaide.edu.au
www.maths.adelaide.edu.au/~droberts





From rrosebru@mta.ca Tue Sep 20 08:15:07 2005 -0300
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Date: Mon, 19 Sep 2005 14:33:50 -0400 (EDT)
From: Richard Blute <rblute@mathstat.uottawa.ca>
To: categories@mta.ca
Subject: categories: Final Octoberfest Announcement
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Hi everyone,

This is a brief final reminder for Octoberfest 2005, to be held October
22nd and 23rd at the University of Ottawa. If you have not already
booked a room, you should do so right away. Also we will be finalizing
the speaker's list this week. So we will need a title and abstract
by Friday. Since we are receiving many submissions, this deadline will
be strict.

All details can be found at the Octberfest website at
http://aix1.uottawa.ca/~scpsg/Octoberfest05/Octoberfest.final1.htm


We look forward to seeing you soon,
Rick Blute
Phil Scott


-- 






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From: "Ronald  Brown" <ronnie@ll319dg.fsnet.co.uk>
To: <categories@mta.ca>
Subject: categories: lax crossed modules
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reply to here

Vaughan, David

An interesting question!

It raises several possible red herrings.

1) What is a lax action of a group (or groupoid) G on a group (or groupoid)
A? There is a paper by Brylinski in Cahier on this, with applications to
K-theory, if I remember rightly.

Another interpretation of this seems to be as a Schreier cocycle (factor
set). A relevant paper is

97. (with T. PORTER), ``On the Schreier theory of non-abelian
extensions: generalisations and computations''. {\em Proceedings
Royal Irish Academy} 96A (1996) 213-227.

It is a useful exercise (which I meant to write down, but ...) to translate
Brylinski into the terms of a map of a free crossed resolution, and so put
this into nonabelian cohomology terms, and potentially allow for calculation
using a small free crossed resoution when possible ....

This suggests what might be  a lax action, but does not complete in an
obvious way into a lax crossed module.

2) Another way is to go to 2-crossed modules (Daniel Conduche), which brings
in relations with simplicial groups (Tim Porter) and higher Peiffer
elements. See also the relations with braided crossed modules and other
things in
59.  (with N.D. GILBERT), ``Algebraic models of 3-types and
automorphism  structures for crossed modules'', {\em Proc. London
Math. Soc.} (3) 59 (1989)  51-73.

3) There are equivalences of categories

crossed modules of groupoids  <--> 2-groupoids   <--> double groupoids with
connections
<--> double groupoids with thin elements.

I have long found the cubical approach easier to follow and to use than the
globular, but it turns out one needs also the globular to define commutative
cubes in cubical omega-categories with connections (see a recent paper by
Philip Higgins in TAC).

This raises the question of "lax cubical omega-categories with connections".
What do you laxify, to get an equivalence with one or other notion of weak
globular (or other?) omega-category??!!

Quite an amusing step, and more do-able,  would be to generalise Gray
categories to: cubical omega-categories C with an algebra structure C
\otimes C \to C, generalising Brown-Gilbert, and using the monoidal closed
structure given in

116. (with F.A. AL-AGL and R. STEINER), `Multiple categories: the
equivalence between a globular and cubical approach', Advances in
Mathematics, 170 (2002) 71-118

A nice point about such algebra structures is that they allow for a failure
of the interchange law, with a measure of that failure, similar to  the way
2-crossed modules give a measure of the failure of the Peiffer law for a
crossed module by using a map { , }: P_1 \times P_1 \to P_2. Is this related
to Sjoerd Crans' teisi?

I have a gut feeling that these strengthened sesquicategories (with a
*measure* of the failure of the interchange law) will crop up in a variety
of situations, e.g. in rewriting, 2-dimensional holonomy, ...., since the
interchange law makes things too abelian, sometimes.

This brings in automorphism structures for crossed modules, I guess
(Brown-Gilbert again, and of course derived from JHC Whitehead, who first
studied such automorphisms).

Another thought: the non abelian tensor product of groups derives from
properties of the commutator map on groups. Why not develop the
corresponding theory for the Peiffer commutator map?

Hope that helps

Ronnie

-------------------------------------------------
*Date:*     Mon, 19 Sep 2005 09:41:44 -0700
*From:*     Vaughan Pratt <pratt@cs.stanford.edu>

*To:*   Ronnie Brown <mas010@bangor.ac.uk>

*Reply-to:*     pratt@cs.stanford.edu

*Subject:*      [Fwd: categories: Question re lax crossed modules]

I'd be interested in knowing this too, in particular what the geometric
significance of laxness is.  Presumably laxness only enters in the
passage from pre-crossed to crossed.

Vaughan

-------- Original Message --------
Subject: categories: Question re lax crossed modules
Date: Mon, 19 Sep 2005 12:44:30 +0930
From: David Roberts <droberts@maths.adelaide.edu.au>
To: categories@mta.ca

I have been looking at categorical groups a little and was wondering
what a lax crossed module is. A search through various databases has
turned up nothing. It would seem that they should be like crossed
modules but only satisfy a weakened equivariance property.

Any pointers toward a definition would be great.


------------------------------------------------------------------------
--
David Roberts
School of Mathematical Sciences
University of Adelaide SA 5005
------------------------------------------------------------------------
--
droberts@maths.adelaide.edu.au
www.maths.adelaide.edu.au/~droberts





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Dear All,
There is an interesting old result on fibrations that may shed light on
this.  It is well known  that if $F\to E\to B$ is a fibration of pointed
spaces then $\pi_1 F \to \pi_1 E$ is a crossed module and in fact any
crossed module can be  given in this form. Now one tries to prove it.
First the action: take a loop   g in E and another a in F. concatentate
to get  gag^{-1}.  This is a loop in E whose image in the base,B, is
null homotopic.  Pick a null homotopy that does this.  Lift it to a
homotopy in E starting at gag^{-1} using the homotopy lifting property
of the fibration.  Evaluate the other end of the lift.  This is a loop
in F. The corresponding element of \pi_1 F is the result of acting on
the class of a by the class of g.
Note the way the action is determined up to homotopy.  The verification
that the rules work up to homotopy is left as an exercise.

I learnt this from a paper by Eric Friedlander, who attributed it to
Deligne.  I suspect it is already essentially in Whitehead's
Combinatorial Homotopy II paper or Peter Hilton's lovely little book on
Homotopy Theory.

It suggests a `homotopy everything' version of crossed module, not just
a lax one.  Its advantage is that it clearly links up the structure with
the quite classical topological version of fibrations and so should be
adaptable to other situations.

Hope this helps.

Tim



David Roberts wrote:

> I have been looking at categorical groups a little and was wondering
> what a lax crossed module is. A search through various databases has
> turned up nothing. It would seem that they should be like crossed
> modules but only satisfy a weakened equivariance property.
>
> Any pointers toward a definition would be great.
>
>
> ------------------------------------------------------------------------
> --
> David Roberts
> School of Mathematical Sciences
> University of Adelaide SA 5005
> ------------------------------------------------------------------------
> --
> droberts@maths.adelaide.edu.au
> www.maths.adelaide.edu.au/~droberts
>
>
>




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From: "Ronald  Brown" <ronnie@ll319dg.fsnet.co.uk>
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Subject: categories: More fancies on lax crossed modules and cubical ideas
Date: Tue, 20 Sep 2005 11:51:30 +0100
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To pursue some ideas suggested by David Robert's queries on lax crossed
modules and Vaughan Pratt's interest:

As said in the previous contribution: crossed modules (over groupoids) are
equivalent to (edge symmetric) double groupoids with connections or thin
structures, and the latter generalise easily to all dimensions. Why use
crossed modules? For the work with Higgins, the aim was  (a) to relate to
classical invariants (relative homotopy groups) , and (b) for calculations.
One thinks of calculation as serial, so the `linear' crossed modules are
appropriate. But for theory, one wants the clear 2-dimensional compositions,
particularly to get `algebraic inverses to subdivisions' for applications to
`local-to-global problems'.

Now multiple groupoids arose from considering the structure held by the
singular cubical complex of a space, SC(X), which in dimension n consists of
maps I^n --> X.  So SC(X) has some claim to be a model of a weak
omega-groupoid. Now for a filtered space X_* one can consider also R(X_*)
which in dimension n is filtered maps I^n --> X_*. This again is a weak
omega-groupoid, at least as much as SC(X) is.

But an advantage is there is Kan fibration p:R(X_*) --> \rho(X_*) where the
latter is a strict omega-groupoid.

32.  (with P.J. HIGGINS), ``Colimit theorems for relative homotopy
groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41.
(except that now we would modify the definition to take homotopies rel
vertices of the cubes, and avoid the J_0-condition, and the theorems still
work).

So this suggests that a `controlled lax omega-groupoid'  R should come
with a Kan fibration R->G where G is a strict omega-groupoid and where R has
lots of lax multiple compositions, [a_{(r)}],  as considered in [32].  (why
not?) This would allow for liftings of multiple compositions from G to R
(loc cit), which should be helpful.  An advantage of cubical over globular
or simplicial is the ease of formulating multiple compositions.

Notice that SC(X) has strict interchange for a 2 x 2 composition, and the
connections have strict transport laws (2 x 2 again) but lax cancellation of
\Gamma^-_i with \Gamma^+_i (in the terms of Al-Agl/Brown/Steiner).

To backtrack a bit:
\mu : M \to P is a crossed module (of groups!) if and only if there is a
pointed fibration F \to E \to B such that \mu is \pi_1 F \to \pi_1 E.
(Loday) The lax version of this is that given such a fibration, then \Omega
F \to \Omega E may be given the structure of lax crossed module, where I
cheat by saying lax means the structure that this has --  `crossed module up
to homotopy'. (I am not sure if this has been written down somewhere!) This
is related to an old paper of Philip R. Heath on, if I remember correctly,
`Groupoid Operations and fibre homotopy equivalences'. Presumably there is
also a recognition principle involved.

This should answer Vaughan's question on the geometry (here topology)
related to lax crossed modules.

Ronnie
www.bangor.ac.uk/r.brown
r.brown@bangor.ac.uk







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Ronnie,

     I don't see these words below but `lax functor' is what came to mind.
As a monoid is a category with one object,
what is the many object version of an ordinary crossed module?

jim



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Subject: categories: Re:  Question re lax crossed modules
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Tim Porter wrote:

> Dear All,
> There is an interesting old result on fibrations that may shed light on
> this.  It is well known  that if $F\to E\to B$ is a fibration of pointed
> spaces then $\pi_1 F \to \pi_1 E$ is a crossed module and in fact any
> crossed module can be  given in this form. Now one tries to prove it.
> First the action: take a loop   g in E and another a in F. concatentate
> to get  gag^{-1}.  This is a loop in E whose image in the base,B, is
> null homotopic.  Pick a null homotopy that does this.  Lift it to a
> homotopy in E starting at gag^{-1} using the homotopy lifting property
> of the fibration.  Evaluate the other end of the lift.  This is a loop
> in F. The corresponding element of \pi_1 F is the result of acting on
> the class of a by the class of g.
> Note the way the action is determined up to homotopy.  The verification
> that the rules work up to homotopy is left as an exercise.
>
> I learnt this from a paper by Eric Friedlander, who attributed it to
> Deligne.  I suspect it is already essentially in Whitehead's
> Combinatorial Homotopy II paper or Peter Hilton's lovely little book on
> Homotopy Theory.
>
> It suggests a `homotopy everything' version of crossed module, not just
> a lax one.


There is a closely related homotopy everything version which I bet can
be adapted to this problem.

Consider \Omega B, the based loop space and F as the fibre over
that base point. Then an argument like that above gives
and action of \Omega B on F which satisfies the usual rules for an
actin only up to homotopy
Easiest way to say it is the adjoint map \MOmega B --> Aut F
meaning the self homotopy equivalences of F
is a strongly homotopy associative map of monids
(if you use Moore loops) or of A_\infty spaces.

It should be in
 "Parallel transport in fibre spaces," Bol. Soc. Mat. Mexicana (1968),
68-86.

or

"Associated fibre spaces," Michigan Math. Journal 15 (1968), 457-470.

Hope that helps

jim

>   Its advantage is that it clearly links up the structure with
> the quite classical topological version of fibrations and so should be
> adaptable to other situations.
>
> Hope this helps.
>
> Tim
>




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Message-ID: <1127237168.43304630dabc2@webmail.u-picardie.fr>
Date: Tue, 20 Sep 2005 19:26:08 +0200
From: Andree Ehresmann <andree.ehresmann@u-picardie.fr>
To: categories@mta.ca
Subject: categories: Conference Charles Ehresmann: 100 ans
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Conference and papers for Charles Ehresmann's centenary

I recall that we organize an International Conference "Charles Ehresmann
100 ans" to commemorate the 100th anniversary of Charles' birth. It will
take place at the Universite de Picardie Jules Verne in Amiens, from
October 7 to 9, 2005. The first day will consist in general lectures on
the life and works of Charles, and their prolongations. The 2 other days,
there will be 2 parallel sessions:

    Category theory, Topology and Geometry, including the 82nd session of
the PSSL "Peripatetic Seminar on Sheaves and Logic" and a session of the
SIC (Seminaire Itinerant de Categories),

    Multidisciplinary applications with a Symposium ECHO V.

The complete program can be found on the Internet site dedicated to
Charles:
http://perso.wanadoo.fr/vbm-ehr/ChEh
which also contains some photos and documents on Charles, and papers dedicated
to him.
The abstracts of the lectures presented to the Conference will appear in Volume
XLVI-3 of the "Cahiers de Topologie et Geometrie Differentielle Categoriques".
The participation to the Conference is free. If you want to participate, please
contact me by sending your name and address to
ehres@u-picardie.fr
Same address if you want to send a paper to be posted on the site.

Sincerely
Andree C. Ehresmann







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Message-ID: <012001c5bece$ca12fe70$2846a8c0@acerorzjm7qpwt>
From: "Urs Schreiber" <Urs.Schreiber@uni-essen.de>
To: <categories@mta.ca>
References: <001d01c5bd6b$adea7020$94a24e51@brown1>
Subject: categories: Re: lax crossed modules
Date: Wed, 21 Sep 2005 19:06:21 +0200
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Ronald Brown wrote, in response to David Roberts:


> I have a gut feeling that these strengthened sesquicategories (with a
> *measure* of the failure of the interchange law) will crop up in a variety
> of situations, e.g. in rewriting, 2-dimensional holonomy, ...., since the
> interchange law makes things too abelian, sometimes.



One can have a 2-holonomy for nonabelian gerbes if a funny condition holds,
called the "fake flatness condition", which is a differential version of the
exchange law, appearing when one realizes a 2-holonomy in a gerbe as a
2-functor from 2-paths to 2-group 2-torsors.

Some people working on bundle gerbes feel that this constraint, which is
derived in the context of strict 2-groups (crossed modules) is "too strong".
While there are straightforward ways to relax conditions in the formalism,
for instance by passing to weak (coherent) structure 2-groups (I guess these
are essentially "the same" as lax crossed modules?) this does not seem to
really address these people's concerns, because after weakening one no
longer deals with Lie groups and Lie algebras, which is what they do.

Hence I'd be extremely interested if somebody came up with a nice weakened
version of crossed modules that would allow to realize 2-holonomy in
non-fake flat gerbes.

Best regards,
Urs Schreiber





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Date: Wed, 21 Sep 2005 13:38:15 -0400 (EDT)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
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To: categories@mta.ca
Subject: categories: Bill on John (and his adequate subcats)
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                 John Isbell's Adequate Subcategories
                            F. W. Lawvere
                   Submitted to the Topology Atlas
               (in response to Mel Henriksen's request)

For mathematicians of my age, the theory of rings of continuous
functions was one of the first exciting research topics we
encountered. Many results of that theory have appeared, but its
ramifications for category theory are still not fully worked out. A
crucial link was provided by John Isbell's contributions around 1960
on the theme of adequate subcategories.

Briefly, a subcategory  A  of a larger category is adequate if every
object  X  of the larger category is canonically the colimit of the
category  A/X  of objects of  A  equipped with structural maps to  X;
John's equivalent definition was that the truncated Yoneda embedding
of the whole category into the category of set-valued contravariant
functors on  A  is actually a full embedding. The following language
is suggestive:

  (a) The objects of  A  are figure-types,
  (b) the objects of  A/X  are particular figures in  X, and
  (c) the morphisms of  A/X  (commutative triangles in the big
      category) are incidence relations between figures.

Thus adequacy means that the large category in question consists of
objects entirely determined by their  A-figures and incidence
relations, and that

  (d)  the morphisms in the whole category are nothing but the
       "geometrically continuous" ones in the sense that they map
       figures to figures without tearing the incidence relations.

For example, if  A  is the category of countable compact spaces then
A  is adequate in many large categories constructed in attempts to
capture the notion of topological space; in this case a morphism can
be identified as a mapping that preserves sequential limits. That
example is one of many illustrating that typical large categories of
mathematics often have quite small adequate subcategories; it had been
studied by Fox in 1945 at the instigation of Hurewicz, who sought a
rational notion of function space for use in algebraic topology and
functional analysis. In fact, for any  A, each space of A-continuous
morphisms has its own cohesion, described again by  A-figures.

The dual notion of a co-adequate subcategory  C  leads to a
contravariant representation of the larger category that can be
described in terms of

  (a)  quantity-types,
  (b)  functions, and
  (c)  algebraic operations on functions.

The dual of the notion of geometric continuity (that is, a name for
naturality of maps of covariant functors instead of contravariant
ones) is

  (d)  "algebraic homomorphism"

These ideas of John Isbell became fused with the conceptions of
Kan, Grothendieck, and Yoneda (emerging in the same period 1958-1960),
to form a basic method of analyzing and constructing mathematical
categories. That method was used in the early 60's by Freyd, Gabriel,
Lawvere, Mitchell, and by Isbell himself, and became as natural as
breathing to many algebraists and topologists during the following
decades.

What does adequacy have to do with rings of continuous functions? The
theory of rings of continuous functions springs from a basic
philosophical hope to the effect that there should be a near-perfect
duality between space and quantity. Such duality questions can be
investigated for a great many different categories, but categorical
considerations suggest that they need to be brought down to earth in
certain respects.

John Isbell was also one of the main developers of the theory of
locales. This theory revealed that the traditional notion of
topological space is algebraic rather than geometric (in the sense of
the above analysis) with the infinitary algebras (frames) of open
sets playing the dominant role; this merely means that the Sierpinski
space, together with "all" its powers, constitutes a coadequate
subcategory of the category of sober spaces. John's insistent quest
for smallness (as a further requirement on co-adequate subcategories
C) brought this analysis qualitatively nearer to real mathematics.

If a small co-adequate subcategory is available, it can often be
reduced to a single object (for example by taking the product of its
objects; or more concretely, the Euclidean plane as a topological
object will often serve the purpose of co-adequacy). The endomorphisms
of that object then parameterize the unary operations whose
preservation by  C-homomorphisms serves to exclude ghosts from among
detected points and figures. Even among those operations a few may be
co-adequate, as the Stone-Weierstrass theorem had shown: addition,
multiplication, and conjugation can replace the monoid of all
continuous operations for that particular task, and there are many
variations on that theme. But apart from such details of presentation,
the implicit insight of Czech and Stone, of Hewitt and Nachbin,
apparently includes smallness of the algebraic theory  C  in terms of
which spaces are to be (co) analyzed.

There was a seeming barrier to the realization of that concrete
insight: set theory, in its striving for larger and larger cardinals,
had neglected to emphasize that all of the cardinals arising in
geometry and analysis in fact satisfy a useful smallness condition:
the category of countable sets is co-adequate in the category of all
small sets. That is essentially John Isbell's formulation; he proved
that it is equivalent to the condition that no small set has the kind
of ghost elements called Ulam measures. John knew full well that his
formulation for abstract sets would imply that many categories having
small adequate subcategories also have small co-adequate
subcategories, thus making possible the desired sort of dualities
between space and quantity.

The ideas of John Isbell contributed to the enlightened understanding
of mathematics by lifting some dark clouds of confusion, and they
continue to be actively developed and diffused.




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Subject: categories: Re: lax crossed modules
To: categories@mta.ca (categories)
Date: Wed, 21 Sep 2005 11:42:55 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>
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Urs Schreiber wrote:

> [...] weak (coherent) structure 2-groups (I guess these
> are essentially "the same" as lax crossed modules?) [...]

Since not everyone will understand this remark by my esteemed
coauthor, let me elaborate.

There's a nice way to weaken the concept of crossed module.
A crossed module is just another way of looking at a group
object in Cat - otherwise known as a "categorical group" or
"strict 2-group".

But, starting with the concept of group object in Cat, one can
then weaken the usual group axioms to natural isomorphisms
and impose suitable coherence laws, obtaining the notion of
"gr-category" or "coherent 2-group".

One could then backtrack and formulate this concept so that it
resembles the concept of crossed module as closely as possible.
I guess this would deserve to be called a "weak crossed module"
or something like that.

All this stuff except the last paragraph is well-known and
summarized here:

John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups,
Theory and Applications of Categories 12 (2004), 423-491.
http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html

One might also seek a "lax" version of the concept of crossed
module, where "lax" is taken in the Australian sense of replacing
equations by morphisms rather than isomorphisms - "lax" as opposed
to "pseudo".  If I were forced to do this, I'd try to do it by
laxifying the concept of group object in Cat.  But, I don't see
which way all the 2-arrows should point.

Best,
jb





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Message-ID: <1127763608.43384e986b4a6@mymail.tcd.ie>
Date: Mon, 26 Sep 2005 20:40:08 +0100
From: Shane O'Conchuir <oconchus@tcd.ie>
To: categories@mta.ca
Subject: categories: Computational category theory paper - request for comments
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Hi all,

I've written a paper on computational category theory but have for a while been
unable to determine whether or not my methods or results are new. I would
greatly appreciate any input from the community on this matter.

The paper, "Proper Diagrams for Constructing Presheaf-Valued Limits", is about
the construction of limits of diagrams of presheaves. The content is not too
deep mathematically and is aimed more at computation. The basic idea is that
given a functor D from a small category C into a category of presheaves, I
construct a second functor D' of the same type where there is a natural
transformation D'->D whose components are monomorphisms. The construction has
the property that the limit of D is isomorphic to the limit of D'. The
intuition is that we remove parts of the presheaves so that the computation of
the limit is easier. I like to think of this approach as orthogonal to the use
of initial functors to construct limits. Both methods preserve the limit but
initial functors reduce the number of objects in the domain of a diagram
whereas our transformation reduces the objects in the codomain of the dia=
gram.

A concise version of the paper is available at
https://www.cs.tcd.ie/Shane.OConchuir/limits/limitspaper.pdf
A draft technical report with the missing proofs and some appendices is
available at
https://www.cs.tcd.ie/Shane.OConchuir/limits/limitstr.pdf
The technical report explains how I derived my definition of "spare" element.

Any comments, corrections, criticisms, or references are welcome. In particular,
 I would like to know if any of this seems familiar (apologies in advance!)
Also, my choice of terminology ("proper", "inconsistent") probably conflicts
with normal use and one of my constructions, 'final proper diagram', may well
be called 'initial proper diagram'.


Many regards,

Shane O'Conchuir
Department of Computer Science
Trinity College Dublin



From rrosebru@mta.ca Mon Sep 26 20:38:45 2005 -0300
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From: "Michael Mislove" <entcsorg@entcs.org>
To: categories@mta.ca
Subject: categories: Announcing www.entcs.org
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Dear Colleagues,

   One of the minor inconveniences that Hurricane Katrina caused has been
the temporary failure of the Tulane email servers, both university-wide
and within the math department.  The latter hosted the ENTCS Macro Home
Page, so progress on publishing ENTCS volumes has been hindered since the
hurricane.

  I am happy to announce that ENTCS now has its own, separate web host,
which can be found at www.entcs.org Please point your browser at this
page, where you will find detailed instructions on how to prepare
proposals for publishing material in ENTCS, as well as instructions about
how to prepare files both for preliminary, hard copy versions of
proceedings for limited distribution at meetings, as well as how to
prepare the final versions of papers for publication online at Science at
Direct. While the ENTCS production has been hampered over the past month
or so, it has been restarted, and publication of ENTCS issues and volumes
is now proceeding as usual, with minimal delays.

  An important point to note is that the procedure for submitting final
versions of files for ENTCS has changed. We now ask that the corresponding
author of each paper - including the Preface of each volume - submit an
electronic copy of the signed Copyright Transfer Form that can be found at
the ENTCS Macro Home Page archives. To do this, authors should download
the pdf file containing the form, print it out and complete it, including
signing it, and then scan it in and include this scanned copy with the
files for the final version of their paper that are sent to the Guest
Editors of their proceedings. This will expedite the publishing of ENTCS
volumes: we now can have final versions online within four weeks of when
the files are sent to me for final processing.

  As usual, if you have any problems or questions about the ENTCS macros,
or about ENTCS in general, please let me know.

  Best regards,
  Mike Mislove

-- 
Michael Mislove
Managing Editor
ENTCS






From rrosebru@mta.ca Tue Sep 27 20:21:33 2005 -0300
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To: categories@mta.ca
Subject: categories: ENTCS
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From: Paul Taylor <pt@cs.man.ac.uk>
Date: Tue, 27 Sep 2005 10:42:34 +0100
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I am sure that I speak for all of us in the category theory
community when I say that I am very (relieved and) pleased to
hear from Mike Mislove that he and ENTCS are alive and well
following the hurricane.

On the other hand, I am alarmed by the keenness of Mike (and
the many conference organisers who use the services of ENTCS)
to give away their colleagues' intellectual property to a company
whose sole contribution nowadays to the process of publication
is to run a web server.

In point of fact, Elsevier DOES NOT require ENTCS authors to
transfer copyright.

When they were processing my paper for last year's CTCS, they
asked me for a copyright transfer, but I refused.  Instead,
they promptly offered me a license agreement.  Although the
first version of this was also unacceptable, I succeeded in
negotiating another one, which I signed, and they published
my paper.

You can obtain a LaTeX version of this agreement from
	www.cs.man.ac.uk/~pt/drafts/Elsevier-licence.tex

I urge all of my colleagues to use this, and not give a
commercial publisher the monopoly of research in theoretical
computer science.  I fully understand that younger researchers
in particular feel that they have no alternative but to give
in to pressure, being in a polically weak position myself.
But I have demonstrated that even individuals can have an
effect, simply by saying no.   If more of us say no then
we will succeed in recovering what is properly ours.

Paul Taylor

PS there is lots of new stuff on the ASD web page since my
last posting to "categories" about it.



From rrosebru@mta.ca Tue Sep 27 20:21:33 2005 -0300
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Date: Tue, 27 Sep 2005 08:49:45 -0400 (EDT)
From: Susan Niefield <niefiels@union.edu>
To: categories@mta.ca
Subject: categories: Union College Conference
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UNION COLLEGE MATHEMATICS CONFERENCE

Saturday and Sunday                          Schenectady
December 3-4, 2005                           New York

This is a preliminary announcement of the twelfth Union College
Mathematics Conference.  This year the conference topics will be algebraic
topology, category theory, and commutative algebra.

In addition to plenary lectures, of interest to the entire conference
audience, there will be shorter contributed talks in parallel sessions.
Anyone interested in giving such a talk should contact one of the
organizers.  The deadline for submission of abstracts is November 11th,
and for registration is November 18th.

PLENARY SPEAKERS

    John Baez (University of California, Riverside)
    David Cox (Amherst College)
    Jesper Grodal (University of Chicago)

The meeting will begin with an evening reception on Friday, December 2,
and end on Sunday afternoon.  For more information about the conference,
including registration, submission of abstracts, housing and
transportation, please visit our website at

    www.math.union.edu/~leshk/05Conference/

Union College is centrally located in New York's capital district about 10
miles from the Albany International Airport, easily accessible by train
from NYC, and just 3 to 4 hours by car from NYC, Boston, and Montreal.

CONFERENCE ORGANIZERS

Category Theory
     Susan Niefield        niefiels@union.edu

Algebraic Topology
     Brenda Johnson        johnsonb@union.edu
     Kathryn Lesh          leshk@union.edu

Commutative Algebra
     Pedro Teixeira        teixeirp@union.edu
     David Vella           vellad@union.edu

We hope to see you in December!




From rrosebru@mta.ca Wed Sep 28 11:36:54 2005 -0300
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Subject: categories: Re: ENTCS
To: categories@mta.ca (categories)
Date: Tue, 27 Sep 2005 23:21:27 -0700 (PDT)
From: "John Baez" <baez@math.ucr.edu>
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Paul Taylor writes:

> In point of fact, Elsevier DOES NOT require ENTCS authors to
> transfer copyright.
>
> When they were processing my paper for last year's CTCS, they
> asked me for a copyright transfer, but I refused.  Instead,
> they promptly offered me a license agreement.  Although the
> first version of this was also unacceptable, I succeeded in
> negotiating another one, which I signed, and they published
> my paper.

When I was an editor for an Elsevier-edited journal -
before I decided they were so nasty I should resign -
I discovered how this works.

They're scared to death that people will switch to free electronic
journals.  But, they don't want to openly cave in and institute a
universal policy of letting authors keep the copyrights to their
work.   So, they cut a deal with anyone who pressures them, but
don't advertise this.

For more on the evils of Reed Elsevier, and what to do about it,
see this:

http://math.ucr.edu/home/baez/journals.html

To see the copyright policies of most publishers, see this:

http://www.sherpa.ac.uk/romeo.php

Best,
jb





From rrosebru@mta.ca Fri Sep 30 09:51:59 2005 -0300
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Date: Thu, 29 Sep 2005 18:15:54 -0400 (EDT)
From: Michael Barr <mbarr@math.mcgill.ca>
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To: Categories list <categories@mta.ca>
Subject: categories: Name of concept?
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Is there a name for the following situation:  I have a diagram of
categories and functors
                 DO(F)
          DO(A) ------> DO(B)
           |             |
           |             |
         H |             |H
           |             |
           v     F       v
           A ----------> B
It does not commute, nor is there even a 2 cell in either direction.
What I do have is illustrated below:
                DO(A)                               DO(A)
                /|\                                 /|\
               / | \                               / | \
              /  |  \                             /  |  \
            H/   |   \DO(F)                     H/   |   \DO(F)
            /    P    \                         /    P'   \
           /     |     \                       /     |     \
          v      |      v                     v      |      v
          A  <== | ==>  DO(B)                 A  ==> | <==  DO(B)
           \     |     /                       \     |     /
            \    |    /                         \    |    /
            F\   |   /H                         F\   |   /H
              \  |  /                             \  |  /
               \ | /                               \ | /
                vvv                                 vvv
                 B                                   B

and, moreover,
           P -------> HF
           |          |
           |          |
           |          |
           |          |
           |          |
           v          v
           FH ------> P'
commutes.





From rrosebru@mta.ca Sun Oct  2 15:59:37 2005 -0300
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Date: Fri, 30 Sep 2005 14:37:14 -0400 (EDT)
From: Michael Barr <mbarr@math.mcgill.ca>
To: Categories list <categories@mta.ca>
Subject: categories: Re: Name of concept?
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Incidentally, did you know that if Z and Z' are defined so that
         a      d      a'
0 --> Z ---> C ---> C ---> Z' ---> 0
is exact, then the homology is the image (= coimage) of a'.a: Z --> Z'?
This is a triviality, but it gives a symmetric definition of homology.
Notice that it defines something even when d.d is not 0.  I guess it is Z
mod Z meet ker(d).

Mike




