Subject: Re: question about distributive categories
From: koslowj@math.ksu.edu (Juergen Koslowski)
Date: Mon, 2 Aug 93 14:29:26 CDT

Vaughan Pratt's reply confirms my suspicion that for a symmetric monoidal
closed category with tensor @ and exponential -o the Keisli category for
the (strong) monad induced by the functor T that maps X to (X -o A) -o A 
(A fixed) is highly unlikely to be cartesian closed AND non-trivial. 

This probably is a well-known result. I'd appreciate pointers to the 
literature.        
-- 
J"urgen Koslowski         | If I don't see you no more in this world
                          | I meet you in the next world
                          | and don't be late!
koslowj@math.ksu.edu      |                         Jimi Hendrix (Voodoo 
Chile)
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: On calculuses of fractions
Date: Tue, 3 Aug 93 12:37:58 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Can anyone supply a reference to the fact that if you add to the 
hypotheses of a calculus of right fractions the assumption that
if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then
so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude
that if the original category has all limits, so does the fraction
category and the canonical functor to the fraction category
preserves them.

Michael Barr
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: New book: Theory and Formal Methods 1993
Date: Tue, 03 Aug 93 14:55:18 +0100
From: Mark Ryan <mdr@doc.ic.ac.uk>

Announcing:

"Theory and Formal Methods 1993"

Eds. Geoffrey Burn, Simon Gay and Mark Ryan

A new title in Springer's "Workshops in Computing" Series.

Proceedings of the First Imperial College Department of Computing
Workshop on Theory and Formal Methods, Isle of Thorns Conference
Centre, Chelwood Gate, Sussex, UK, 29--31 Match 1993.

The Theory and Formal Methods Section of the Imperial College
Department of College has an international reputation for research
into the foundations of computer science, and the application of this
theory to real computing problems. In March 1993 it held the first in
a proposed series of workshops on theory and formal methods at the
Isle of Theory Conference Centre in Sussex, UK. This volume contains
revised versions of the papers presented at the workshop. They cover
four main areas --- semantics, concurrency, logic, and specification
-- with several papers spanning a variety of disciplines. The
contributions fall into two main categories: review papers which
provide the reader with an introduction to some specific areas being
studied by the Section, and research papers which give details of the
latest results in these areas.

THEORY AND FORMAL METHODS 1993 provides a comprehensive overview of
the work being carried out by one of the world's leading research
centres in theory and formal methods. It will be of interest to
practitioners and researchers, as well as post- and undergraduate
students.


OVERVIEW PAPERS

Geoffrey Burn
The Abstract Interpretation of Functional Languages
 
Roy Crole
Deriving Category Theory from Type Theory
 
Steve Vickers
Geometric Logic in Computer Science

Chris Hankin
Graph Rewriting Systems and Abstract Interpretation



RESEARCH PAPERS

Samson Abramsky
Interaction Categories
 
Mark Dawson
Animating LU

Abbas Edalat
Dynamical Systems, Measures and Fractals via Domain Theory

Abbas Edalat
Self-Duality, Minimal Invariant Objects and Karoubi
Invariance in Information Categories

Lindsay Errington, Chris Hankin and Thomas Jensen
Reasoning about GAMMA Programs

Jose  Fiadeiro and Tom Maibaum
Generalising Interpretation between Theories in the Context 
of (pi-)Institutions

Simon Gay and Raja Nagarajan
Modelling SIGNAL in Interaction Categories

Reinhold Heckmann
Product Operations in Strong Monads

Michael Huth
On the Equivalence of State-transition Systems

Stuart Kent
Towards a Modal Logic of Durative Actions 

Marta Kwiatkowska
Concurrency, Fairness, and Logical Complexity

Marta Kwiatkowska and Iain Phillips
Concurrency and Conflict in CSP

Sarah Liebert
A Complete Axiom System for CCS with a Stability Operator

Ian Mackie, Leopoldo Roman and Samson Abramsky
An Internal Language for Autonomous Categories

Juarez Muylaert Filho and Geoffrey Burn
Continuation-passing Transformation and Abstract Interpretation

Iain Phillips
A Note on the Expressiveness of Process Algebra

Mark Ryan
Prioritising Preference Relations

David Sands
Laws of Parallel Synchronised Termination

Zvi Schreiber
Implementing Process Calculi in C

Paul Taylor
An Exact Interpretation of While

Irek Ulidowski
Congruences for tau-respecting Formats of Rules


Theory and Formal Methods 1993
Eds. Geoffrey Burn, Simon Gay and Mark Ryan
Springer-Verlag London, 1993.
336pp (approx), 32 figures, 12 tables. Soft cover. Publication due:
September 1993. Price: 37 pounds (30 pounds to members of the BCS).
ISBN 3-540-19842-3.


ORDER FORM

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+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Lie?
From: Kirill Mackenzie <K.Mackenzie@sheffield.ac.uk>
Date: 4 Aug 93 14:13:48 BST

This is in response to Andre Joyal's message of Jul 13, and in enlargement
of Ross Street's of last week. I have been away for the last three weeks.

As Ross already intimated, quite a lot of work has been done on the
infinitesimal theory associated with Lie groupoids since Pradines' notes of
the 1960s. I just add a few comments.

(1) The object which Joyal describes:

>               It is a pair (A,D) where A is a commutative
> R-algebra and D is a Lie algebra (over R) such that
>
> 1) D is acting on A (as derivations):
>    X(fg) = X(f)g + fX(g) , [X,Y](f) = X(Y(f)) - Y(X(f))
> 2) D is equipped with an A-module structure such that
>         (fX)(g) = fX(g)    and  [X,fY] = X(f)Y + f[X,Y]
>   (for any f,g in A and X,Y in D)

was defined abstractly by Herz in 1953 and called a Lie pseudo-algebra. It
has since been reinvented by a very large number of people, most of whom
also invented a new name; there are some 14 different terminologies in the
literature.

Pradines was unique in defining the narrower concept of Lie algebroid: a
vector bundle $A$ on base $M$ (the set of identities when the Lie algebroid
comes from a Lie groupoid) together with a vector bundle morphism
$a\colon A \to TM$ and a bracket $[\ ,\ ]$ of global sections of $A$ which
makes the real vector space of global sections into a real Lie algebra and
obeys $a[X,Y] = [aX,aY]$ and $[X,fY] = f[X,Y] + a(X)(f)Y$ for global sections
$X,Y$ of $A$ and smooth functions $f$ on $M$. This is the object which
matters for the Lie theory of Lie groupoids. Note that the partiality
of the multiplication for groupoids gets transformed into the module
structure for the global sections and the map $a$, rather than into a
"partial bracket".

It can be interesting to think of the associated Lie pseudo-algebra as an
infinite-dimensional Lie algebra associated to the infinite-dimensional
group of admissible sections (H in Joyal's notation). But so far as I know
there is no sufficently developed general theory that one could then
apply. Rather, the Lie groupoid structure enables one to do directly in this
case (most of) what one wishes a general theory of infinite-dimensional Lie
groups could do. (For a groupoid which comes from a principal bundle, the
group H of admissible sections is the group of gauge transformations.)

The Lie groupoid/Lie algebroid formalism allows one (amongst many other
things) to pretend that any manifold is a Lie group, any (sufficiently
smooth) equivalence relation arises from a smooth Lie group action, and
so on.


(2) Pradines' notes of the 1960s sketched a Lie theory for Lie groupoids
and Lie algebroids, but gave very few details. In the 1987 book which Ross
mentioned I gave a full account for locally trivial Lie groupoids and
transitive Lie algebroids (= the map $a\colon A\to TM$ is surjective).
In this case, one can use methods quite different from those which Pradines
sketched.

In the general case quite a lot is known but there is no systematic account
and some things are still open. Since my book was written Weinstein and,
independently, Karasev, have created an extensive theory of symplectic
groupoids, and special techniques are available in this case also. Symplectic
groupoids are very far from being locally trivial.

(Terminology: In my book, a Lie groupoid is always taken to be locally
trivial, and the general concept is called a differentiable groupoid.)



(3) Functoriality can be achieved without any dualization. Philip Higgins and
myself (J. Alg. 129, 1990, 194--230) developed a means of working with
general morphisms of Lie algebroids. The basic categorical constructions with
groupoids (equivalence between actions and action morphisms, semi-direct
products, quotients, etc) then also hold for abstract Lie algebroids.

It is a curious fact that, although there is a very natural definition of
morphism for Lie pseudo-algebras, it does not correspond to the infinitesimal
map induced by a morphism of Lie groupoids. This is another reason for
distinguishing between Lie algebroids and Lie pseudo-algebras.

The dual of a Lie algebroid has a natural Poisson structure, extending the
linear Poisson structure on the dual of a Lie algebra (Courant, Trans AMS,
319, 1990, 631--661). One knows that a linear map between Lie algebras is a
morphism iff its dual is a Poisson map. Extending this to Lie algebroids runs
into the problem of how to dualize a base-changing morphism of vector
bundles. Higgins and I found a way around this ("Duality for base-changing
morphisms...", Math. Proc. Camb. Phil. Soc. to appear very shortly).



(4)

> It is easy to see that we have a contravariant functor
>
>       {Lie groupoids}--->{de Rham complexes}
>
> and it follows from the work of Pradine that it has a left adjoint Exp
> (the exponential functor)
>
>      Exp:{de Rham complexes}--->{Lie groupoids}
>> when it is restricted to complexes (A,F,D) in which A is an algebra of
> smooth functions on a manifold.
>
> The functor Exp is full and faithful. Its image should consist
> of some kind of simply connected Lie groupoids. I guess (I should really
> read Pradine ...) these groupoids are those for which the fibers of d0 are
> simply connected. ??

Not all Lie algebroids arise as the Lie algebroids of Lie groupoids, even
in the transitive case. Counterexamples were given by Almeida and Molino
(CRAS (Paris), 300, 1985, 13--15). For transitive Lie algebroids I gave a
cohomological obstruction to integrability; this is a kind of nonabelian first
Chern class. See Chapter 5 of the book already mentioned for the case where
the base is simply-connected and Cahiers 28, 1987, 29--52, for the general
case. A general framework for thinking of this obstruction is given in JPAA
58, 1989, 181--208.


(5) There is also a Lie theory associated with double Lie groupoids. Here
a double groupoid is a groupoid object in the category of groupoids: it is
a double Lie groupoid if it has a smooth structure making all four
groupoid structures Lie and such that the map which sends any square to (say)
its right side and its bottom side is a surjective submersion. Double Lie
groupoids arise by natural games with ordinary groupoids, and in homotopy
theory, but also in the integration of Poisson Lie groups (Lu and Weinstein,
CRAS (Paris), 309, 1989, 951--954) and Poisson groupoids.

Taking the infinitesimal object associated to a double Lie groupoid is a
two-step process; the first step yields an LA-groupoid, that is, a groupoid
object in the category of Lie algebroids. These are of interest in themselves:
the cotangent bundle of a Poisson Lie groupoid is an LA-groupoid, for example.
See Adv. Math. 94, 1992, 180--239.

QUESTION: What is the infinitesimal object corresponding to a (Lie)
2-category? Is there a Lie theory here also?



Lastly, to revert to Jim Stasheff's original question, given all the above
and given that there is also a theory of Lie semigroups
(Hilgert-Hofmann-Lawson), it should be straightforward to define an
infinitesimal object associated to a "Lie category". How far one could get
with a Lie theory is another matter entirely. My impression of the semigroup
theory is that the absence of inverses causes plenty of difficulties
there already.

Kirill Mackenzie
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: On calculuses of fractions
Date: Thu, 5 Aug 93 10:10:44 +1000
From: kelly_m@maths.su.oz.au (Max Kelly)

Michael Barr asks, in a letter dated 3 Aug,

``Can anyone supply a reference to the fact that if you add to the 
hypotheses of a calculus of right fractions the assumption that
if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then
so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude
that if the original category has all limits, so does the fraction
category and the canonical functor to the fraction category
preserves them."

For closely-related results, see 

Kelly, Lack, & Walters, Coinverters and categories of fractions for
categories with structure, Applied Categorical Structures, to appear
in first issue; and a paper (ibid) by Kelly and Lack to which this
appeals.

Max Kelly.



+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: On calculuses of fractions
Date: Thu, 5 Aug 93 08:53:57 +1000
From: street@macadam.mpce.mq.edu.au

>Can anyone supply a reference to the fact that if you add to the 
>hypotheses of a calculus of right fractions the assumption that
>if {s_i: X_i --> Y_i} is a family of arrows, all in Sigma, then
>so is \prod s_i: \prod X_i --> \prod Y_i, then you can conclude
>that if the original category has all limits, so does the fraction
>category and the canonical functor to the fraction category
>preserves them.

A relevant reference in the case of finite products is

Brian Day, Note on monoidal localisation, Bulletin Australian Math Soc 
Volume 8 (1973) 1 - 16.

Brian called a class S of arrows s in a monoidal category  V a monoidal
class when s * X and X * s are both in the class when s is. Then
the localisation V[S^-1] becomes monoidal with tensor-product preserving
projection.

The result you state was surely known to Brian and others (such as 
Harvey Wolff) and may even occur in Brian's work, but I don't have time 
now to scan his work. I'll ask him when next I talk to him.

Regards,
Ross

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: ?lie
Date: Thu, 5 Aug 93 15:54 GMT
From: MAS010@BANGOR.AC.UK (Ronnie Brown)


\documentstyle{article}

\begin{document}

This is a further point in reply to the report on Lie groupoids
by Andre Joyal. 



Pradines' groupoid analogue of the functor {Lie
algebras}$\rightarrow$ {simply connected Lie groups} has 
subtleties which do not appear in the group case. These are
stated in his CR Note of 1966, and an outline of the
constructions was explained by him to me in the years from 1981. 

1)Holonomy: From a Lie algebroid there may be obtained a locally
Lie groupoid, i.e. a groupoid $G$ with a Lie structure on a
subset $W$  of $G$ containing the identities of $G$. However,
unlike the group case, the simple conditions which arise
naturally for this situation do not imply that the Lie structure
extends to give a Lie groupoid structure on $G$. Instead, under
suitable conditions, a new groupoid, the holonomy groupoid
$Hol(G,W)$ is obtained which has a Lie structure and which maps
to $G$. This groupoid is the minimal "covering" of $(G,W)$ which
has a Lie structure. Complete statements and proofs for the
topological case are given in Aof and Brown, "The holonomy
groupoid of a locally topological groupoid" Top. Appl. 47 (1992) 
97-113. 
(This is essentially an account of Th\'eor\`eme 1 of Pradines
Note.)

2) The natural analogue of a simply connected toplogical group
is a topological groupoid whose stars (the inverse images of the
source map) are simply connected. This we call star simply
connected. The problem is then to construct from a Lie groupoid
$G$ a star simply connected Lie groupoid $\tilde{G}$ and
morphism $p:G\rightarrow  \tilde{G}$ which is a universal
covering morphism on stars. The existence of this is a part of
the statement of Th'eor\`eme 2 of the same Note. Full details of
the construction and full prooofs of its properties are in a
Bangor preprint 93.09, R Brown and O Mucuk, ``The monodromy
groupoid of a Lie groupoid".  This also discusses the Lie case
of holonomy (in line with Pradines' Note). The point is that is not 
hard to construct a groupoid which is the universal cover of each 
star; the problem is to get a topology making it a Lie groupoid. The 
proof given follows Pradines' outline (given verbally) in using 
holonomy arguments.  

3) For applications to foliations, one needs to recognise that a
foliation on a paracompact manifold gives rise to a locally Lie
groupoid. This is proved in Bangor preprint 93.10 by Brown and
Mucuk. 

These two preprints have not yet been duplicated, but I can send a TEX 
file of 93.09, a postscript file of 93.10 (3MB)  (this has some 
pictures), or hard copies to anyone interested. 
These results formed a part of Mucuk's thesis (Jan, 1993). 



The locally trivial case of the monodromy construction is dealt
with in Mackenzie's book ``Lie groupoids and Lie algebroids in
differential geometry" (Cambridge, 1987), by a different method. 

Kock and Moerdijk also have work on related ideas for local equivalence 
relations. 

\end{document}



Ronnie Brown

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: xypic
Date: Fri, 6 Aug 93 13:58:53 -0400
From: cfw2@po.CWRU.Edu (Charles F. Wells)

Has anyone successfully used xypic with LaTeX on an MS DOS 
machine?  I am having some very puzzling problems I would like
to discuss with someone, but I didn't want to take up the category
network's time by listing them all to everyone.

--Charles

--
Charles Wells, Department of Mathematics, Case Western Reserve University
10900 Euclid Avenue, Cleveland OH 44106-7058, USA
Phone 216 368 2880 or 216 774 1926
FAX 216 368 5163
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: lie?
Date: Fri, 6 Aug 93 9:35 GMT
From: MAS010@BANGOR.AC.UK 

Damn: I just noticed my note got the map $p: \tilde{G} \to G$ the 
wrong way round, though I am sure the correction was spotted easily.

I should also say that aim of the Brown-Mucuk paper (again following 
Pradines)  is to use methods of free groupoids to get the monodromy 
principle as well: the globalisation of local morphisms to the star 
universal cover. 

The nice point is that if (G,W) is a locally Lie groupoid one gets 
star covering morphisms
\tilde G \to Hol(G,W) \to G,
and there may be interesting Lie groupoids sandwiched between the 
first two. 

There is a problem of terminology. From a topological groupoid G one 
obtains a groupoid $\tilde{G}$ which is the star universal cover. If 
$G = X \times X$, then \tilde G is the fundamental groupoid of X. If G 
is the equivalence relation of a foliation, then \tilde G is the 
socalled ``homotopy groupoid'' of the foliation, i.e. the fundamental 
groupoid of X with the leaf topology. There is a temptation to call 
\tilde G the ``fundamental groupoid of G'', but this conflicts with 
the fundamental groupoid of the underlying space of (the arrows of ) 
G.  

Ronnie
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: compactly generated locales.
From: Francois Lamarche <gfl@doc.ic.ac.uk>
Date: Mon, 9 Aug 93 15:58:29 BST


There is a well-known cartesian closed category of topological spaces, 
described in Mac Lane's CWM. A Hausdorff space is compactly generated if
a set is closed iff its intersection with every compact subspace is closed.

Question: does this construction generalize to an arbitrary topos? In other
words is there a notion of compactly generated locale, such that we get a
cartesian closed full subcategory of the category of locales over that topos?
Ideally all separation axioms should be dropped. Also one would (well I would)
like to have these locales to have enough points.

   Francois Lamarche, Imperial College.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: CTCS-5 Program
Date: Mon, 9 Aug 1993 13:52:27 +0200
From: F.J.de.Vries@cwi.nl


                            CTCS 5

             Category Theory in Computer Science

                    7th-10th September 1993
                CWI, Amsterdam, The Netherlands
                                                                
                 Program and Registration Form


PROGRAM

Tuesday 7th September

 9.15 Opening

 9.30 - 10.30   Saunders Mac Lane (Invited Speaker)
10.30 - 11.00           Tea/Coffee
11.00 - 11.40   Pietro Cenciarelli and Eugenio Moggi
                A Syntactic Approach to Modularity in Denotational
                Semantics
11.40 - 12.20   Claudio Hermida and Bart Jacobs
                Fibrations with Indeterminates: Contextual and
                Functional Completeness for Polymorphic Lambda Calculi 
12.20 - 13.00   G. Michele Pinna and Axel Poign\'e
                The Mathematics of Event Automata

13.00 - 14.00           Lunch

14.00 - 14.40   A.J. Power
                Why Tricategories?
14.40 - 15.20   N. Sabadini, R.F.C. Walters and Henry Weld
                On Distributive Automata and Asynchronous Circuits
15.20 - 16.10           Tea/Coffee
16.10 - 17.00   Alain Prout\'e
                ``Substitution Should Not Respect Equality''
17.00 - 17.40   Adam Obtulowicz
                Graphical Sketches, a Finite Presentation of
                Infinite Graphs
18.00 - 19.30   Reception

Wednesday 8th September

 9.30 - 10.30   N. Shanin (Invited Speaker)
                An Finitary Version of Mathematical Analysis
                Oriented to Computer Science
10.30 - 11.00           Tea/Coffee
11.00 - 11.40   Paul Taylor
                Intuitionistic Ordinals and Tarski's Theorem
11.40 - 12.20   Marcelo P. Fiore
                Cpo Categories of Partial Maps

12.30                   Conference Outing 

18.30                   Conference Dinner

Thursday 9th September

 9.30 - 10.30   G. Rosolini (Invited Speaker)
10.30 - 11.00           Tea/Coffee
11.00 - 11.40   Daniele Turi and Bart Jacobs
                On final Semantics for Applicative and
                Non-deterministic  Languages
11.40 - 12.20   S. Soloviev
                Reductions in Intuitionistic Linear Logic
12.20 - 13.00   B.P. Hilken and D.E. Rydeheard
                A Theory of Classes: Proofs and Models

13.00 - 14.00           Lunch

14.00 - 14.40   A. Carboni and P. Johnstone
                Connected Limits, Familial Representability and
                Artin Glueing
                
14.40 - 15.20   B.P. Hilken and D.E. Rydeheard
                Computing Colimits
15.20 - 16.10           Tea/Coffee
16.10 - 17.00   John G. Stell
                Sesqui-Categories and their Applications to
                Rewriting Systems
17.00 - 17.40   S. Kasangian, G. Mauri and N. Sabadini 
                (presented by Sebastiano Vigna) 
                Trees of Traces: A Categorical View

Friday 10th September

 9.30 - 10.30   A. Joyal (Invited Speaker)
10.30 - 11.00           Tea/Coffee
11.00 - 11.40   Akira Mori and Yoshihiro Matsumoto
                Unification in Categories and Proof Search in
                Intuitionistic Propositional Calculus 
11.40 - 12.20   Martin Hofmann
                Sound and Complete Axiomatisations of Call by Value
                Control Operators
12.20 - 13.00   Eike Ritter and Valeria de Paiva
                Syntactic Multicategories and Categorical
                Combinators for Linear Logic
13.00 - 14.00           End/Lunch


PROGRAM AND ORGANIZING COMMITTEE
S. Abramsky, P.-L. Curien, P. Dybjer, G. Longo, G. Mints, J.
Mitchell, E. Moggi, D. Pitt, A. Pitts, A. Poigne, D. Rydeheard, 
F.J. de Vries and E. Wagner.

LOCAL ARRANGEMENTS
Fer-Jan de Vries
Department of Software Technology,
CWI,
Kruislaan 413,
1098 SJ Amsterdam,
The Netherlands

CONFERENCE ADMINISTRATOR
CTCS-5, c/o CWI
Ms. Anna Baanders
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Fax. +31-20-5924199
email: anna@cwi.nl

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CONFERENCE FEE
The conference fee for CTCS-5 is NLG 500 for advance registration
(to be paid by 10 August 1993) and NLG 600 after 10 August 1993. 
The fee includes admission to all sessions, lunches and coffee 
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+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Group photograph
Date:        Mon, 09 Aug 93 16:38:58 ADT
From: es@math.mcgill.ca (Elaine Swan)


Can anyone please supply me with the
correct email address for Dr. Chiment?
The one I have, <jjcl@cornell.edu> seems to
be incorrect. This request is for Prof. J.
Lambek.
Thank you
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: New Textbook
Date: Tue, 10 Aug 1993 10:30:51 +0100 (BST)
From: Roy Crole <rlc@doc.ic.ac.uk>



          PRELIMINARY ANNOUNCEMENT              
-------------------------------


The following book  will be available shortly:


               CATEGORIES      FOR      TYPES

Cambridge Mathematical Textbooks, Cambridge University
Press


Roy L. Crole,   Imperial College, University of London.


Abstract:

This textbook explains the basic principles of
categorical type theory and illustrates some of the
techniques used to derive categorical semantics for
specific type theories.  It introduces the reader to
ordered set theory, lattices and domains, and this
material provides plenty of examples for an
introduction to category theory.  Categories, functors
and natural transformations are covered, along with the
Yoneda Lemma, cartesian closed categories, limits and
colimits, adjunctions and indexed categories.  Four
kinds of formal system are presented in detail, namely
algebraic, functional, second order polymorphic and
higher order polymorphic type theories. For each of
these type theories a categorical semantics is derived
from first principles, and soundness and completeness
results are proved.  Correspondences between the type
theories and appropriate categorical structures are
formulated, along with a discussion of internal
languages.  Specific examples of categorical models are
given, and in the case of polymorphism both PER and
domain-theoretic structures are considered.
Categorical gluing is used to prove results about type
theories.

Aimed at advanced undergraduates and beginning
graduates, this book will be of interest to theoretical
computer scientists, logicians, and mathematicians
specialising in category theory.


Contents:

(1) Order, Lattices and Domains    [1--36]
(2) A Primer on Category Theory    [37--119]
(3) Algebraic Type Theory     [120--153]
(4) Functional Type Theory     [154--196]
(5) Polymorphic Functional Type Theory     [197--268]
(6) Higher Order Polymorphism     [269--308]
Bibliography      [309--313]
Index         [314--335]


ISBN 0521 450926 (HB)

The Edinburgh Building, Shaftesbury Road, CAMBRIDGE,
CB2 2RU,  England,  UK.

40 West 20th Street, New York, NY 10011-4211, USA.

10 Stamford Road, Oakleigh, Victoria 3166, Australia.










+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: change of address
Date: Thu, 12 Aug 1993 09:53:10 -0400
From:   Reinhard.Boerger@FERNUNI-HAGEN.DE

Please send all further messages for me to Hagen, where I returned
after six months at York. My internet address is:

      Reinhard.Boerger@Fernuni-Hagen.de

My postal address is:

        Fachbereich Mathematik
        Fernuniversit"at
        Postfach 940
        58084 Hagen
        Germany

Please note the new postal code according to the new German system.
I should appreciate if the other network participants in Germany
also posted their new postal codes by the network. Greetings
                                 Reinhard B"orger
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: categories and complexity
Date: Thu, 12 Aug 93 09:31:06 EDT
From: otto@triples.Math.McGill.CA (James Otto)


                     Kalmar, Linear Space, and P
                               J. Otto
                      Department of Mathematics
                          McGill University
                         otto@math.mcgill.ca
                           August 12, 1993

is available by anonymous ftp from triples.math.mcgill.ca.  It is in
uuencoded compressed postscript in pub/otto/0-k-ls-p.uu.  Thus

ftp
ftp> open triples.math.mcgill.ca
ftp> Name (...): anonymous
... Guest login ok, send e-mail address as password.
Password:
ftp> cd pub
ftp> cd otto
ftp> get 0-k-ls-p.uu
ftp> quit
uudecode 0-k-ls-p.uu
uncompress 0-k-ls-p.ps.Z

We use 2-simplices to cut down the primitive recursive functions to
the Kalmar elementary functions, and 1-simplices to cut down the
primitive recursive functions to the functions of complexities linear
space and P time.  This translates work of [Leivant Marion],
[Bellantoni], and [Bellantoni Cook].  As 2-simplices are less
degenerate than 1-simplices, we first consider Kalmar elementary.
Then, successively modifying characterizations, we consider linear
space and P time.

Remarks.

1.  Kalmar elementary and linear space are levels 3 and 2 of the
Grzegorczyk Hierarchy.  E.g. see [Rose].

2.  The characterizations can be viewed as programming languages whose
typing guarantees, without explicit bounds checking, that programs
represent precisely the functions of the complexity class.

Most of the category theory we need is in [Barr Wells].  For the
little that we need of 2-categories, e.g. see [Kelly Street], [Makkai
Pare].

This paper revises and expands the April 18, 1993 paper `Kalmar
Elementary and 2-Simplices'.  In particular, besides minor changes and
corrections,

1.  Linear time is omitted, as the characterization I had translated
may need some repair.

2. Machine based soundness and completeness proofs for the linear
space and P time characterizations are added.

The machine based proofs unfold [Ritchie], [Cobham] from the proofs of
[Bellantoni], [Bellantoni Cook] and make this paper largely self
contained.  [Bloch] has related work on machines.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Lie?
Date: Thu, 12 Aug 1993 16:59:28 -0400
From: chase@math.cornell.edu (Stephen Chase)

I would like to comment on the considerations raised in the recent 
messages of Joyal, Street, Mackenzie, and Brown, but from a somewhat
different perspective.

Although I accept the fact that the notion of Lie algebroid is what matters
for the Lie theory of Lie groupoids, I think there are a couple of reasons why 
the general concept of Lie pseudo-algebra should not be dismissed.

The first is that examples arise from a variety of sources in algebra.  A
fundamental one comes from any  k-algebra  A (k  a field, say) with 
commutative subalgebra   K.  Namely, let  A+  be the "differential
normalizer" of  K  in  A:  The set of all  u  in  A  with  ux - xu  in  K
for all  x  in  K.  Then  (K,A+)  is a Lie pseudo-algebra over  k; moreover,
for fixed  K/k, the functor  A ------> (K,A+)  has a left adjoint
(K,L) ------> U(K.L), the universal enveloping algebra of a Lie pseudo-
algebra constructed by Rinehart [TAMS 108 (1962)].  (Actually, this is not
quite the whole story, since in general one must consider algebra maps
K ----> A  that are not necessarily embeddings).  Another important
example arises from a Lie  k-algebra  L  acting on a commutative  k-algebra
K  by derivations:  (K,K#L)  is then a Lie pseudo-algebra, where  K#L  is 
the tensor product of  K  with  L  over  k  with a suitable twisted
bracket operation.

My other reason to keep Lie pseudo-algebras in mind is that they should
form part of a larger theory of groupoid schemes and formal groupoids.
Although some years ago I explored and used some special cases of these
notions (see below), I am not aware of any systematic development of them
in the literature.  That is, with the exception of the appendix on
groupoid schemes and Hopf algebroids in Ravenel's  1986 book on stable
homotopy.  ;I would be interested in hearing of other references.  (In this
regard I should also mention Huebschmann's paper on Poisson algebras
[Crelle 408 (1990)], in w;hich Lie pseudo-algebras play an important role).

As is suggested by the examples discussed above, Lie pseudo-algebras have
a long history in the Galois theory and Brauer group literature.  Jacobson's
Galois correspondence for purely inseparable extensions of exponent one
is between intermediate fields in such an extension and restricted Lie
pseudo-subalgebras of  (K,L), with  L  the restricted Lie pseudo-algebra of
k-derivations of  K [Amer. J. Math. 66 (1944)].  Hochschild classified
Brauer classes of central simple  k-algebras split by  K (with  K/k  as
above) in terms of extensions of restricted Lie pseudo-algebras [TAMS 76
(1954) and 79 (1955)].  In my paper [Amer. J. Math 98 (1976)] I generalized
Jacobson's theorem to modular purely inseparable extensions of arbitrary
exponent by a method that rests on the notion of the group scheme  G#  of
admissible sections of a finite groupoid scheme.  Although at the time
it seemed clear to me that an analogous theory of Lie groupoids should exist,
until 1988 I was unaware of the work that had been done in that area, and I
am indebted to Ronnie Brown for informing me of it.  It is not surprising
that the group of admissible sections should play a role in Galois theory,
since it provides a very natural definition of the automorphism group  (or
"group of symmetries") of a collection of objects, in a manner which takes
into account not only the symnmetries of the individual objects but also
their isomorphisms with each other.

The literature described above also suggests the appropriate groupoid
analogue of the cocommutative Hopf algebra that arises as the continuous
dual of the algebra of representative functions of a formal group, or the
Hopf-Sweedler dual of the algebra of representative functions of a group
scheme.  In my view, this analogue should be Sweedler's notion of a
xK-bialgebra [Groups of simple algebras, Publ. IHES no. 44 (1975)] (although
presumably with some sort of antipode).  To oversimplify a bit, a  
xK-bialgebra
is a  k-algebra  A  which is also a  K-coalgebra, with suitable axioms
linking the two structures (the module of primitive elements of  A  then
yields a Lie pseudo-algebra for  K/k).  These algebras are a ubiquitous as
Lie pseudo-algebras:  In addition to the universal enveloping algebras
mentioned above, they include the classical smash product  K#G (G  a group
acting on  K  by  k-algebra automorphisms) and the algebra of differential
operators on  K.  But logically (if not historically) the paradigm here is
the  k-algebra   A = kC  of a small category  C, with  K  the  k-algebra of
k-valued functions on the set  X  of objects of  C; the coalgebra structure
on  A  is defined in a manner entirely analogous to that of a group 
algebra (it also satisfies a similar universal property).  Finally, the
grouplike elements of  A  constitute the monoid of admissible sections of  C.

                                                        Steve Chase

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: noncomm geom/n-cats and quantization
Date: Sat, 14 Aug 93 08:38:19 -0400
From: jds@math.upenn.edu

In cae you haven't noticed, Penkov's review of Manin's book in July Bull AMS
says:
Manin explains also why monoidal categories can be viewed as a unification 
base
(at least) of quantum geom and supergeom.

p.108
jim stasheff
+++++++++++++++++++++++++++++++
Date: Sat, 14 Aug 93 08:40:20 -0400
From: jds@math.upenn.edu

Dan Freed (dafr@math.utexas.edu)
has a preprint
Higher algebraic structures and quantization
where the higher alg structures are in fact n-cats or n-ple cats
or whichever variant is appropriate
jim
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Diaconescu for sheaves
Date: Fri, 13 Aug 1993 14:20:00 +0000
From: sjv@doc.ic.ac.uk (Steven Vickers)

Does anyone know where there's a proof of a relativized version of the
well-known fact that the topos of sheaves over a locale classifies the
points of the locale? What I imagine is something like the following:

Let E be a topos, and A a frame in it. Let E' be the topos of internal (in
E in some suitable sense) sheaves over the locale of A. Then for any topos
F there is an equivalence between -

* the category of geometric morphisms from F to E'

and

* the category whose objects are pairs (f, x) where f: F -> E is a
geometric morphism and x: A -> f_*(Omega_F) is a frame homomorphism (with
suitable morphisms between these pairs).

(Presumably this could be deduced from a more general theorem that
generalizes Diaconescu by dealing with toposes of sheaves over sites, not
just presheaves over categories.)

I feel a bit stupid, because I know topos theorists take results like this
for granted all the time. But I can't track down any account of the
details, nor quite convince myself that methods such as Joyal and Tierney's
("pretend E is ordinary sets but take care to reason constructively") do
the trick in this case.

Steve Vickers.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: UACT Participants List-posting corrected
Date: 18 Aug 93 01:15:43 EDT
From: "Paul H. Palmquist" <76600.1050@CompuServe.COM>

Another correction to the UACT address list.
My Zip Code got mashed into my email address.
The correction follows.

Paul H. Palmquist
13112 Dewey St
Los Angeles, CA 90066

Email:   76600.1050@compuserve.com

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: change of address
Date: Thu, 19 Aug 93 19:02:50 BST
From: Thorsten Altenkirch <alti@dcs.ed.ac.uk>

Reinhard Boerger recently wrote:

   Please note the new postal code according to the new German system.
   I should appreciate if the other network participants in Germany
   also posted their new postal codes by the network. Greetings

There are some servers set up which you can use to find out the new
german zipcodes via the network. The easiest way is to do

        rlogin -l plz plz.isr.uni-stuttgart.de

You are connected to a program which translates addresses. Unluckily
the help text is in german. However the usage is simply to type in an
address as in the example followed by control-D.

I would appreciate it if the noise on the network created by change of
address mails could be kept on a minimum level. If there is serious
interest in having addresses available somebody should collect them
centrally and make them available by ftp (like Vaughan Pratt's
structdir). Another way to distribute address changes is to make it
available to the finger program by putting the new address into the
.project file (e.g. try "finger alti@yell.dcs.ed.ac.uk").

Cheers,
Thorsten

 Thorsten Altenkirch            Kennst du das Land, wo die Zitronen blu"hn,
 Laboratory for Foundations     Im dunkeln Laub die Gold-Orangen glu"hn,
 of Computer Science            Ein sanfter Wind vom blauen Himmel weht,
                                Die Myrte still und hoch der Lorbeer steht,
 University of Edinburgh        Kennst du es wohl?      Dahin! Dahin
                                Mo"cht ich mit dir, o mein Geliebter, ziehn.
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: McGill Octoberfest
Date:        Sun, 22 Aug 93 19:42:24 ADT
From:        fox@triples.math.mcgill.ca


                CATEGORY THEORY MEETING:  OCT 9-10, 1993

            CATEGORY THEORY RESEARCH CENTER, MCGILL UNIVERSITY


Dear Colleague,

     We look forward to seeing you again this fall.  We will meet
in the basement of Burnside Hall (805 Sherbrooke St W) at 8:30
Saturday morning for coffee, and the first talk will be at 9:00.
If you wish to speak, please contact Michael Barr as soon as possible.
A final schedule will be drawn up Saturday morning.

        In the past these meetings have been run without registration fees.
However, the combination of increasing costs and decreasing grants has
made it impossible to continue on this basis.  In order to recover part
of our costs, this year there will be a registration fee of $25 for
professors, $15 for students.

     Below you will find a list of hotels and tourist rooms within
easy walking distance of McGill.  To obtain the quoted price you should
mention McGill when making your reservation.  The area code for Montreal
is 514.  If you have any further questions, contact Tom Fox.

                                Hotels:

L'Appartement, 455 Sherbrooke W, 284-3634, $72
Howard Johnson Plaza, 475 Sherbrooke W, 842-3961, $79
Citadelle, 410 Sherbrooke W, 844-8851, $79
Four Seasons, 1050 Sherbrooke W, 284-1110, $120
Holiday Inn, 420 Sherbrooke W, 842-6111, $94
Journey's End, 3440 Park Ave, 849-1413, $79-88
Hotel du Parc, 3625 Park Ave, 288-6666, $90
Versailles*, 1659 Sherbrooke W, 933-3611, $89

                          Tourist Rooms:

Ambrose, 3422 Stanley, 288-6922, $45-50
Armor*, 151 Sherbrooke E, 285-0140, $55-62
Casa Bella, 258 Sherbrooke W, 849-2777, $35-70
Pierre*, 169 Sherbrooke E, 288-8519, $35-55

*20 minute walk from McGill


Dept of Mathematics and Statistics
McGill University
805 Sherbrooke West
Montreal, Quebec
CANADA  H3A 2K6

Michael Barr  barr@triples.math.mcgill.ca
Tom Fox       fox@triples.math.mcgill.ca
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Lie?
From: Kirill Mackenzie <K.Mackenzie@sheffield.ac.uk>
Date: 17 Aug 93 16:50:54 BST

>  Date: Thu, 12 Aug 1993 16:59:28 -0400
>  From: chase@math.cornell.edu (Stephen Chase)

>  Although I accept the fact that the notion of Lie algebroid is what matters
>  for the Lie theory of Lie groupoids, I think there are a couple of reasons
>  why the general concept of Lie pseudo-algebra should not be dismissed.

I certainly agree that Lie pseudo-algebras deserve independent study.
I have a survey article "Generalized Lie theories: Lie algebroids and Lie
pseudo-algebras as algebraic invariants in differential geometry" in
near-final stage, with a bibliog of about 100 references; this is available to
people who have an interest in the subject. The orientation is towards the
use of Lie algebroids and Lie pseudo-algebras as a unifying concept in
first-order differential geometry, but the bibliog is intended to be complete
and covers purely algebraic work that I know of.

>                                                    Another important
>  example arises from a Lie  k-algebra  L  acting on a commutative  k-algebra
>  K  by derivations:  (K,K#L)  is then a Lie pseudo-algebra, where  K#L  is
>  the tensor product of  K  with  L  over  k  with a suitable twisted
>  bracket operation.

>                     That is, with the exception of the appendix on
> groupoid schemes and Hopf algebroids in Ravenel's  1986 book on stable
> homotopy.

If one considers Lie algebroids over a fixed base manifold, or if one
considers Lie pseudo-algebras over a fixed (commutative) algebra, then
everything in both cases is quite similar to the case of finite-dimensional
Lie algebras over a field, at least as regards basic definitions and
constructions. If one allows arbitrary base manifolds or base algebras
things become very different.

Firstly, morphisms of Lie algebroids and morphisms of Lie pseudo-algebras
no longer correspond: the concept of morphism of Lie algebroid which arises
by differentiating a morphism of Lie groupoids does not corrrespond to
the natural concept of morphism of Lie pseudo-algebra. Higgins and I
(Math. Proc. Camb. Phil. Soc., to appear) defined concepts of comorphism,
so that a comorphism of Lie algebroids induces a morphism of their Lie
pseudo-algebras, and a morphism of Lie algebroids induces a comorphism of
Lie pseudo-algebras; this in particular enables a duality to be defined in
both categories.

From this perspective, the point of the # construction is that it enables
general Lie pseudo-algebra morphisms to be reduced to the algebra-preserving
case. In (K,K#L) above, L itself can be a Lie pseudo-algebra. Now if L --> L'
is a morphism of Lie pseudo-algebras over a morphism of k-algebras K --> K',
there is an action of L on K', and the morphism can be lifted to (K',K'#L),
and is now a morphism over K'. [MPCPS, cited above.] In the Lie algebroid
context the analogue of the # construction was found independently and called
an "action Lie algebroid"; it is the infinitesimal analogue of the Ehresmann
construction of a groupoid from an action of a group(oid) and its
characterization in terms of covering morphisms. See J. Alg. 129, 1990,
194--230 and refs there.

Incidentally, morphisms of Lie pseudo-algebras also arise in Poisson geometry:
a Poisson morphism induces not a morphism of the cotangent Lie algebroids but
a morphism of the Lie pseudo-algebras of 1-forms.

Quotients of Lie algebroids in the base-varying case are more complicated
and need a kind of reduction process. I am not sure what the situation is
with quotients of Lie pseudo-algebras. See J. Alg. 129, 1990, 194--230 again.
The corresponding general quotient for (Lie) groupoids [Jour LMS, 42, 1990,
101--110] should have implications for Ravenel's 1986 Appendix.

Kirill Mackenzie
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject:  
From: Oege de Moor <oege@ipl.t.u-tokyo.ac.jp>
Date: Wed, 25 Aug 93 11:25:00 JST


Let F be an endofunctor on a category C that has finite
products. A strength of F is a natural transformation

    phi(A,B) : FA x B  -> F(A x B)

satisfying the obvious coherence conditions with respect 
to the terminal object and associativity of products. Is 
anything known about uniqueness of such strengths, if C 
is not well-pointed?

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Notice of Workshop
Date: Tue, 24 Aug 93 21:22:11 EST
From: La Monte H Yarroll <piggy@hilbert.maths.utas.edu.au>

Forwarded message:

>From H.GUSTAFSON@qut.edu.au Tue Aug 24 14:05:16 1993
Date: Tue, 24 Aug 93 13:49 +1000


******************************************************************************
**

                     3C WORKSHOP 93
         CATEGORIES, COMPUTING AND COMBINATORICS
                                   
                   September 1-3, 1993

       School of Computer Science and Engineering
             University of New South Wales  

 * Second announcement and Programme *



This is the second in a series of _informal_ workshops 
exploring the connections between category theory, logic
theoretical computer science and algebraic combinatorics.
It will provide a forum at which experts and interested
workers from outside the field can exchange ideas.

The scope of the workshop has been significantly widened.
There will be talks in the following general areas:

1.  Categorical logic and categorical models of computation
    (e.g. imperative programming, distributed computing)

2.  The role of category theory in computer system design
    (e.g. object-oriented design, information systems and
    database theory, design process automation)

3.  Applications of category theory to combinatorics and
    combinatorial topology

4.  Algebraic combinatorics

Our emphasis will be on the connections between different
strands of research, and on new ways of thinking about
old problems.  We hope to promote cross-fertilisation
between areas.

There will be a series of tutorials on each of these
areas during the afternoon of Wednesday, September 1;
these will assume only a passing acquaintance with
category theory and/or combinatorics.  They are meant
to provide a suitable background for the more
specialised talks on Thursday and Friday.

We expect a number of international visitors to be
participating: these will include Takayuki Hibi
(Hokkaido; algebraic combinatorics).  
Arrangements for other visitors have not yet been 
finalised, but they _tentatively_ include Nicoletta 
Sabadini (Milan), Lin Ying Ming(Szechuan), 
Eric Wagner (IBM Yorktown Heights) and Rod Burstall 
(Edinburgh).

Anyone is welcome to participate.  We intend the talks
to be accessible to people without an extensive background
in category theory and/or combinatorics; one of our major
goals is to make the most recent ideas and results in these
areas available to a wider community.  In particular we
would hope to have a number of industrial participants, as
we did last year.  Research students are also especially
welcome. 

The programme will be finalised in August.  Prospective
speakers should contact Amitavo Islam as soon as possible.
A limited amount of financial assistance is available to
assist speakers travelling from interstate.

Organisers:   Wesley Phoa (UNSW)           general organiser

              Karl Wehrhahn (Sydney)       combinatorics
              Dominic Verity (Macquarie)   category theory

              Amitavo Islam (Sydney)       administration
              Linda Milne (UNSW)           UNSW arrangements

This workshop is being run in conjunction with the Software
Engineering Research Group (UNSW), the Sydney Category Group,
the CATACOMB Group (Sydney) and the Theory Group (Macquarie).

******************************************************************

         WORKSHOP ON CATEGORIES, COMPUTING AND COMBINATORICS
                                   
              School of Computer Science and Engineering
                    University of New South Wales
                                   
                   Meeting Room 1, Samuels Building
                         September 1-3, 1993
                                   
                         TENTATIVE PROGRAMME

WEDNESDAY, September 1 -- tutorials

        1:00    "Combinatorial Species" (Bill Unger)

        2:00    "Ehrhart Polynomials" (Takayuki Hibi)

        3:00    [coffee]

        3:30    "Natural transformations and coherence" (Nick Verne)

        4:30    "Exactness for datatypes" (Barry Jay)

THURSDAY, September 2

        9:00    "Graphs, Hall Algebras and quantum groups" (Jie Du)

        9:55    [coffee]

        10:15   "Toric varieties in combinatorics" (Vladimir Popov)

        11:15   "Cyclotomic identities in combinatorics" (Adrian Nelson)

        12:10   [lunch]

        1:30    "Insight into information system structures using
                category theory" (Kit Dampney and Mike Johnson)

        2:30    "Consistency for SML: an operational perspective"
                (Ed Kazmierczak)

        3:25    [coffee]

        3:45    "A categorical description and justification for a
                synthesizer" (Trudy Weibel)

        4:45    "The programming language dr" (Desmond Fearnley-Sander)

FRIDAY, 3 September

        9:00    "Rewrite systems and 2-categories" (Ross Street)

        9:55    [coffee]

        10:15   "Distributive categories and parallel computation"
                (Henry Weld, Bob Walters and Nicoletta Sabadini)

        11:15   "Categorical term rewriting" (Barry Jay)

        12:10   [lunch]

        1:30    "There is an anti-intuitionistic co-implication
                operator in Set^op" (La Monte Yarroll)

        2:30    "Survey of Face Vectors and Ehrhart polynomials of convex 
polytopes"
                (Takayuki Hibi)

        3:25    [coffee]

        3:45    "Survey of Fibonacci lattices" (Rowan Kemp)

        4:45    "Categories, dependent types and donkey sentences"
                (Wesley Phoa)
 
******************************************************************************
******

This letter was sent to us by Karl Wehrhahn whose e-mail address is:  

wehrhahn_k@maths.su.oz.au


It has been forwarded by Helen Gustafson,  CMSA Membership Secretary

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re:  Uniqueness of strengths
Date: Wed, 25 Aug 93 13:41:30 -0600
From: robin@cpsc.ucalgary.ca (Robin Cockett)



In partial answer to Oege de Moor's query concerning the uniqueness of 
strength in non-well-pointed finite product categories 
                             ... there is an, admittedly, special case 
when the strength is unique: namely when the type F is a strong initial 
(or final) datatype.  The strength can be described as a fold (or unfold) 
and the uniqueness of this guarantees the uniqueness of the strength 
(actually this is dependent on the uniqueness of the strengths of the 
functors over which the datatype is defined ... which inductively one 
assumes are datatypes).

Strong datatypes were described in "Strong Catgeorical Datatypes I" by 
myself and Dwight Spencer (p 141-169, Category Theory 1991) -  also 
see Dwights thesis.  More recently Bart Jacobs has given an alternative 
treatment which elides much of the fibrational and 2-categorical aspects 
present in our treatment.

There are besides various other circumstances which force strengths to be 
unique.  E.g. if F is cartesian over a functor G with a unique 
strength (that is there is a strong natural transformation F --> G with all 
naturality squares pullbacks) then its strength is uniquely determined. 
These observations were made in "Data III" (with Dwight: an unpublished 
follow on to the yet unpublished "Data II"!!) and were recently sharpened 
when Barry Jay visited here (Calgary).

BUT does anyone have an example of a functor with a non-unique strength?

-robin

(Robin Cockett)
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Beck's definition of module
Date: Mon, 30 Aug 93 12:42:02 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

In his 1967 thesis (but actually essentially completed by 1964) Beck 
defined an A-module, for an object A of a category _A_ to be an
abelian group object in the slice _A_/A.  This turned out to mean
2-sided A-modules, left A-modules, left A-modules and left A-modules
in the categories of associative algebras, commutative (associative)
algebras, groups and Lie algebras, resp., that is in all cases
the desired coefficients for cohomology.  This is all widely known
among categorists.  Has a full exposition of this ever been published
and, if so, where?

--Michael
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Beck's definition of module
Date: Tue, 31 Aug 1993 08:57:54 -0400 (EDT)
From: MTHFWL@ubvms.cc.buffalo.edu

I too would like such an exposition. Hopefully it will also explain why
modules may have tensor products. Also what about the dual formulation
for say distributive categories, in particular explaining why the
infinitesimal neighborhoods of the diagonal " can't be defined" for
arbitrary objects in the gros Zariski topos.       Bill
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: october fest
Date: Tue, 31 Aug 93 09:33:10 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

In the past, most people who wanted to talk at the meeting told me about
it at the last minute.  Increasingly, people have asked in advance for a
slot.  Unless I have lost some, I have so far 11 requests.  There is a
practical limit of about 15, since we have to finish early enough on
Sunday to allow people to return home.  Below are the requests I have so
far, names, email addresses, titles or descriptions and abstracts or any
other relevant information on the talks.  If you have made a request, be
sure to check if you are on the list.  If you want to request a slot, do
not tarry.  The order below is in the order I received them, BTW.
 
Michael
 
 
 
Giulio Katis <katis_p@maths.su.oz.au>
working on Cauchy completion
 
 
Jonathan Smith <jdhsmith@pollux.math.iastate.edu>
Duality for semilattice representations (with A. Romanowska)
 
We present general machinery for extending a duality between complete,
cocomplete concrete categories to a duality between corresponding
categories of semilattice representations.  This enables known dualities
to be regularised.  Among the applications, regularised Lindenbaum-Tarski
duality shows that the weak extension of Boolean logic (i.e. the semantics
of PASCAL-like programming languages) is the logic for semilattice-ordered
systems of sets.  Another application enlarges Pontryagin duality by
regularising it to obtain duality for commutative inverse Clifford monoids.
 
 
Till Plewe <>
on when a locale product of metrizable spaces is spatial
 
 
Rick Blute <RBLUTE@acadvm1.uottawa.ca>
Contextual Logic (joint with Robert Seely and Robin Cockett)
 
 
Andreas Blass <ablass@math.lsa.umich.edu>
TBA
 
 
Djordje Cubric <cubric@triples.math.mcgill.ca>
Interpolation property for bicartesian closed categories
 
 
Bob Gordon <gordon@euclid.math.temple.edu>
Enrichment Through Variation (joint with John Power)
 
 
L Gaunce Lewis Jr <gaunce@ichthus.syr.edu>
a talk about the equivariant Freudenthal suspension theorem
 
One of those nice situations when just a little touch of category theory
cleans up a mess in topology.
 
 
Richard Wood <rjwood@cs.dal.ca>
Distributive adjoint strings
 
 
Stacy Finkelstein <stacy@saul.cis.upenn.edu>
TBA
 
 
 
Robin Cockett <robin@cpsc.ucalgary.ca>
Copy Categories.
 
These are symmetric monoidal categories in which every object has a natural
coassociative cocommutative comultiplication -- but no (natural) counit.
Examples include the category of partial maps of a finitely complete
category, the Kleisli category of the exception monad of a distributive
category, ...
I shall describe the category of "formal propositions" of a copy category
and why this gives insight into the embedding of a distributive
category into an extensive category (its the 2-category theory behind it!)
 


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