Subject: Re: Beck's definition of module
Date: Thu, 2 Sep 93 07:02:50 -0700
From: rowan@garnet.berkeley.edu

I have been using modules of universal algebras in my work, and plan
to continue doing so.  The following papers of mine use the concept:

        Enveloping ringoids of universal algebras.  Dissertation,
        University Microfilms International, 1992.  The modules of a
        universal algebra A are the same as the left modules of the
        enveloping ringoid Z[A] of the universal algebra.  An appendix
        treats modules in detail, including restriction and induction
        functors, the centralizer of a module, and some important examples.

        Enveloping ringoids.  To appear in Algebra Universalis in Day
        conference special issue.  Summary of thesis.

        Maximal subalgebras of CM algebras.  In review.  Classifies
        maximal subalgebras of CM algebras (e.g. groups, rings, Lie algebras,
        lattices) into 7 types.

I was disappointed recently when the concept, which I feel is very important,
was never mentioned at the July UACT conference at MSRI.  (Perhaps if
Dr. Barr had attended...)

                                        William H. Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Beck's definition of module
Date: Fri, 03 Sep 93 16:09:11 +0100
From: Valeria de Paiva <Valeria.Paiva@cl.cam.ac.uk>

Jon Beck told me that his thesis was never published and that he
wrote quite a bit about the "modules" that never appeared even in
the thesis. I guess if someone wanted to publish it as a technical
report of some description, Beck might be persuaded to give them the
manuscript. But this is off the top of my head, one would have to
contact him and ask.

Unfortunately I couldn't even try do this in Cambridge, as only the
Computer Laboratory produces technical reports...

Valeria de Paiva
------------------------------------------------------------------------------
Valeria de Paiva,                  |
University of Cambridge            | Phone: +44 (0)223 334418
Computer Laboratory                | Fax: +44 (0)223 334678
New Museums Site, Pembroke Street  | JANET: Valeria.Paiva@uk.ac.cam.cl
Cambridge CB2 3QG, England, UK     | Internet: Valeria.Paiva@cl.cam.ac.uk
------------------------------------------------------------------------------
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: About an Outline (1)
Date: Fri, 3 Sep 93 18:37:55 +0200
From: lair@frunip62.bitnet (Christian Lair)

     COMMENTAIRES SUR UN  " OUTLINE "  (1)



C. Wells a recemment rendu public (et disponible par FTP, en
ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte
intitule:
     " Sketches : Outline with References ".
La version multigraphiee, du 6 juillet 1993, qui m'est parvenue
necessite quelques commentaires ... En voici un premier.



La PREMIERE esquisse des categories - completement DETAILLEE
et en termes tout a fait EXPLICITES - se trouve en [Ehresmann, 1966].
Il est donc pour le moins SURPRENANT que l'Outline affirme
peremptoirement (en son point 4.2, lignes 2 et 3) que:
     " The ideas, not in sketch language, date back to Lawvere [1966] ".
Et il est tout a fait INSUFFISANT que l'Outline se limite a signaler
innocemment (en son point 4.2, lignes 1 et 2) que:
     " The sketch for categories is given in detail in [Barr and Wells,
      1990] ... and in [Coppey and Lair, 1988] ... ".



REFERENCES (autres que celles figurant dans le Outline )

[Ehresmann, 1966] : C. Ehresmann, Introduction to the theory of
     structured categories, Techn. Rep. 10, Univ. of Kansas, Lawrence
     (1966).



Christian LAIR
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: *** Publication Announcement ***
Date: Tue, 7 Sep 93 19:05:21 +1000
From: domv@macadam.mpce.mq.edu.au (Dominic Verity)

The following work has recently been re-printed as Macquarie University
Mathematics Report no. 93-123:

                Enriched Categories, Internal Categories and
                             Change of Base.

                            By Dominic Verity
                        
           (Doctoral Thesis, prepared under the supervision of 
            Dr. Martin Hyland, submitted to Cambridge University
            in April 1992 and sucessfully examined in July 1992)

Copies may be obtained by sending a request to me at the following address:

        domv@macadam.mpce.mq.edu.au

A brief description follows:

Part I: Change of Base.
-----------------------

As the utility of generalised category theories has become more apparent, a
coherent description of what happens when we change our base category has
become more and more important. For instance suppose we have two monoidal
categories V and W and a homomorphism H:V--->W which admits a right
adjoint, then we might ask for a full description of the structures that H
induces between the bicategories of V- and W- enriched categories and
functors (or profunctors). A closely related problem asks for a description
of the analogous structures induced on bicategories of internal categories,
fibrations or stacks by a geometric morphism of toposes f:E--->F.

Some partial results in this direction are well known, for instance in all
of the situations described above we will get biadjoints between the
bicategories of generalised categories and functors. In general the action
of base change on bicategories of profunctors (aka bimodules) is more
difficult to describe. Various "local adjointness" notions, of differing
utility and complexity, have been introduced to cope with this problem, but
none of these have been fully satisfactory. A significant difficulty with
this approach is that a given morphism of bicategories may admit many
different local adjoints.

Here we give a complete solution to this problem by considering base change
structures between  bicategories of functors and profunctors at one in the
same time. To do this we consider proarrow equipments (M,K,*) (in the sense
of Wood [4]), which we show are the objects of a family of closely related
"bicategory enriched" categories, called EHom, EMor, EcoMor and EMap. These
are bicategory enrichment in the sense that they are enriched in the strict
traditional way over the closed category of (small) bicategories and
normalised homomorphisms, wherein B^C is the bicategory of normalised
homomorphisms, strong transformations and modifications form C to B.

Just as in a 2-category we may describe adjunctions equationally in terms
of unit and counit 2-cells, we may interpret analogously the notion of
(normalised) 'biadjunction' in such bicategory enriched categories. It
turns out that base change gives rise to biadjoints of this type in the
enriched categories of proarrow equipments introduced above. This fact is
proved in detail for all of the (closely related) examples above,
culminating in a "comparison lemma" for equipments of stacks. An important
application of this framework is a precise general result about the effect
of base change on the weighted colimits (or limits) that a generalised
category possesses. This follows directly from the description of weighted
limits in terms of kan extensions/liftings of profunctors and
representability.

It is worth pointing out that this approach to base change is very close in
spirit to that of Carboni, Kelly and Wood [2] (wherein all bicategories are
locally ordered) which it generalises.

Part II: Double Limits.
-----------------------

At the International Category Theory meeting at Bangor North Wales in 1989,
Bob Pare introduced an alternative approach to describing limits in
2-categories which used Double Categories rather than 2-weights (cf [3]).
He went on to describe a particularly nice class of well behaved 2-limits,
which he dubbed "Persistent" and defined in terms of stability properties
with respect to equivalences. These he characterised in terms of structural
properties of the double categories that parameterise them. However, while
he was able to construct a double category which parameterised the same
limits as any given 2-weight, he admitted that constructing a 2-weight from
a double category was more problematic.

We might think of this as a change of base problem, from Cat-enriched
category theory to the theory of categories internal to Cat (which is
exactly what double categories are). To make this precise we must first
recognise that coequalisers in Cat are not well behaved, being unstable
under pullback, so we cannot construct a bicategory of profunctors internal
to it. However we can embed Cat in the far better behaved category of
simplicial sets and do our work there. All this we make precise and then
apply the work of part I to relate closed classes of 2-weights to
corresponding classes of double categories satisfying certain closure
properties.

Once all this has been done it becomes a triviality to provide a "basis" of
fundamental double limits which collectively generate the closed class of
persistent limits. In fact products and splitting of idempotents along with
constructions called inserters and equifiers are enough, which demonstrates
that  Pare's persistent limits coincide with the Flexible ones defined (in
the context of 2-weights) by Street and Kelly et al [1]. 


References:

[1] Bird, Kelly, Power and Street, Flexible limits for 2-categories, JPAA
61, November 1989.

[2] Carboni, Kelly and Wood, A 2-categorical approach to change of base and
geometric morphisms I, Cahiers de Top. et Geom. Diff. Categoriques
32(1991), 47-95.

[3] Pare, Double Limits, July 1989, Unpublished notes from Bangor summer
meeting.

[4] Wood, Abstract proarrows I, Cahiers de Top. et Geom. Diff. 23-3(1982),
279-290.


++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: About an Outline (2)
Date: Wed, 8 Sep 93 18:29:53 +0200
From: lair@frunip62.bitnet (Christian Lair)


     COMMENTAIRES SUR UN  " OUTLINE "  (2)



C. Wells a recemment rendu public (et disponible par FTP, en
ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte
intitule:
     " Sketches : Outline with References ".
La version multigraphiee, du 6 juillet 1993, qui m'est parvenue
necessite quelques commentaires ... En voici un deuxieme.



Les PREMIERES esquisses de categories munies de choix de limites et
colimites de "types" (i. e. de formes d'indexations) "varies" se trouvent
en [Lair, 1970] et [Burroni, 1970].
Pour traiter de questions de monadicite que ces premieres esquisses
ne permettaient pas de resoudre directement, d'autres esquisses
pour ces memes categories (mais mieux adaptees) ont ete
construites et tout aussi explicitement detaillees en [Lair, 1977]
et [Lair, 1979] (voir aussi [Lair, 1975]). On y trouve egalement les
esquisses de categories munies d'extensions de Kan a droite et
a gauche de "types" (i. e. le long de foncteurs) "varies".
L'Outline fait donc preuve d'une grande NEGLIGENCE en se limitant
a signaler INGENUMENT (en son point 4.2, lignes 4 a 8) que :
     " Many types of categories with extra-structure ... can be sketched
      by an FL sketch ... These include categories with various types
      of canonically-chosen limits and colimits ... Some examples are
      in [Coppey and Lair, 1988], [Barr and Wells, 1985] ... and [Wells,
      1990] ".



REFERENCES (autres que celles figurant dans le Outline)

[Lair, 1970] : C. Lair, Constructions d'esquisses et transformations
     naturelles generalisees, Esquisses Math. 2, Paris (1970).

[Burroni, 1970] : A. Burroni, Esquisses des categories a limites et des
     quasi-topologies, Esquisses Math. 5, Paris (1970).

[Lair, 1977] : C. Lair : These de Doctorat es Sciences, Amiens (1977).



Christian LAIR
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: My new address
From:   Hongde Hu <hhu@mathstat.yorku.ca>
Date:   Fri, 10 Sep 1993 08:44:36 -0400

My new address is 
     

Hongde Hu
Dept. of Math. and Stat.
York Univ.
4700 keele St.
North York, Ont.
Canada M3J 1P3

e-mail: hhu@mathstat.yorku.ca


Please send any correspondance to the above address.


Hongde Hu

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: About an Outline (3)
Date: Mon, 13 Sep 93 17:26:27 +0200
From: lair@frunip62.bitnet (Christian Lair)

     COMMENTAIRES SUR UN  " OUTLINE "  (3)



C. Wells a recemment rendu public (et disponible par FTP, en
ftp.cwru.edu, repertoire math/wells, fichier sketch.dvi) un texte
intitule:
     " Sketches : Outline with References ".
La version multigraphiee, du 6 juillet 1993, qui m'est parvenue
necessite quelques commentaires ... En voici un troisieme,
concernant l'esquisse des 2-categories, dont on peut se
douter par avance qu'elle ne figure pas qu'en [Power et
Wells, 1992].



En [E. Burroni, 1970] sont etudiees les "categories
discretement structurees" par une categorie (cartesienne) V :
en termes "actuels", elles s'identifient aux categories
enrichies par  V . On y trouve, notamment, la description
detaillee d'une FP-esquisse petite  Ecatenrich(x) de sorte que :
- pour tout ensemble  x  et pour toute categorie cartesienne V ,
la categorie  Mod ( Ecatenrich(x) , V )  est equivalente a la
categorie des V-categories (petites) qui ont  x  pour ensemble
d'objets (si,  E  etant une esquisse et  C  une categorie, on
convient de noter  Mod ( E , C )  la categorie des modeles
de  E  dans  C ).
En particulier, si  V = Cat , la categorie  Mod ( Ecatenrich(x) , Cat )
est donc equivalente a la categorie des 2-categories (petites)
qui ont  x  pour ensemble d'objets.

En [Ehresmann, 1963a et 1963b] sont introduites les (petites)
categories structurees par une categorie (localement petite et
finiment complete) C  : en termes qui se veulent "modernes",
elles s'identifient aux categories internes a  C  . De la sorte,
la categorie des petites categories structurees par  C  est
equivalente a la categorie  Mod ( Ecat , C )  (si  Ecat  designe
l'esquisse des categories).
Par exemple, les categories doubles de [Ehresmann, 1963a et
1963b] sont les categories structurees par  C = Cat  : ainsi,
une categorie double a une  "categorie des objets", une
"categorie des fleches" ... En particulier, les 2-categories
s'identifient donc aux categories doubles dont la "categorie
des objets" est une categorie discrete.

De [Lair, 1975b] ressort (notamment) que la categorie des FP-
esquisses petites est munie d'une structure monoidale,
symetrique et fermee "canonique" (parmi d'autres). De sorte
que le produit tensoriel "canonique"  EoE' , d'une FP-esquisse
petite  E  par une autre FP-esquisse petite  E' , verifie :
- si  C  est une categorie localement petite et finiment complete,
alors les trois categories  Mod ( EoE' , C )  ,  Mod ( E , Mod(E',C) )
et  Mod ( E' , Mod(E,C) )  sont "canoniquement"   quivalentes.
En particulier, si  C = Ens , si  E = Ecatenrich(x)  et si  E' = Ecat ,
alors on voit IMMEDIATEMENT que   Ecatenrich(x) o Ecat   est
une esquisse pour les 2-categories qui ont  x  pour ensemble
d'objets.
De meme,  si  C = Ens  et si  E = E' = Ecat , alors on voit
IMMEDIATEMENT que  Ecat o Ecat  est une esquisse pour les
categories doubles. Comme il est PLUS QUE TRIVIAL de construire
une esquisse  Ecatdis  pour les categories discretes, il
est PLUS QU'ELEMENTAIRE d'en deduire qu'une somme (fibree)
convenable  "au dessus de Ecat " :
          (Ecat o Ecat)      +       Ecatdis
                            Ecat
est une esquisse pour les 2-categories.

L'Oultine est donc PLUS QU'INCONSEQUENT en se contentant de
proclamer NAIVEMENT (en son point 4.2, lignes 3 et 4 ) que:
     " The sketch for 2-categories is given in [Power and Wells,1992]" .



REFERENCES (autres que celles figurant dans le Outline)

[Ehresmann, 1963a] : C. Ehresmann, Categories structurees, Ann. Sc.
     Ec. Norm. Sup., t. 80, pp. 349-426, Paris (1963).

[Ehresmann, 1963b] : C. Ehresmann, Categories doubles et categories
     structurees, Note aux C.R.A.S., t. 256, pp. 1198-1201, Paris (1963).

[E. Burroni, 1970] : E. Burroni, Categories discretement structurees
     et triples,  Esquisses Math. 5, Paris (1970).



Christian LAIR
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Modules mailing list
Date: Mon, 13 Sep 93 18:43:44 -0700
From: rowan@garnet.berkeley.edu

I am thinking of establishing a modules mailing list, and account where
a bibliography of articles on modules (as apparently first defined by
Jon Beck, unless it was folklore before that) could be found.  If anyone is
interested in being on such a mailing list, please send mail to

        rowan@garnet.berkeley.edu

along with any comments.

Best regards, Bill Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: The paper `Outline of Sketches' 
Date: Tue, 14 Sep 93 20:00:39 -0400
From: cfw2@po.CWRU.Edu (Charles F. Wells)



C. Lair has been sending messages to this mailing list with
additions and corrections to my article, "Sketches : Outline
with References", which I prepared for the joint meeting on
universal algebra and category theory in Berkeley last July.
It is available by ftp from ftp.cwru.edu in the math/wells
directory (the file is sketch.dvi).  

The paper is a preliminary version, and in the next few weeks I
intend to post a list of addenda and corrections to the paper in
the same directory.  They will include the papers Lair has
mentioned, although I do not have most of them.  (I expect to
request some of them by interlibrary loan.)  They will also
include some other papers that I am embarrassed at having omitted
because I DO own them.  (These include papers by T. Fox and P.
Johnstone and at least one by C. Lair.)

Perhaps sometime next year I will have time for an extensive
rewrite of the paper.  It has expository shortcomings as well as
bibliographical ones and I would hope to remedy some of them. 
Any suggestions concerning expositions and bibliography will be
welcome.

The paper begins, "This document is an outline of the theory of
sketches with pointers to the literature.  An extensive
bibliography is given." It is NOT A HISTORY OF THE DEVELOPMENT
OF SKETCHES.  I did not claim that it was a history and I do not
intend to make it one.  The main emphasis is to describe papers
in the literature that I think will be usable by those who want
to learn about the subject, including people in universal
algebra, theoretical computer science and other areas who would
perhaps find expository papers and books more usable than the
original sources.

Even so, I want the bibliography to be inclusive and so
appreciate being told of papers not mentioned.  

One specific comment:  Any category theorist with some
experience with sketches could derive the sketch for
2-categories.  The paper by Power and me mentioned by Lair
includes that sketch explicitly because our paper is written for
theoretical computer scientists, who might not find it so easy
to come up with the sketch.

--
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: categories & baseball
Date: Wed, 15 Sep 93 10:36:37 EDT
From: fox@triples.Math.McGill.CA (Thomas F. Fox)

The Montreal Expos are four and a half games out of first.  Why should
category theorists care about this?  Because that means they are
threatening to play baseball in October, which will make it difficult to
find hotel rooms for the CATEGORY THEORY OCTOBERFEST at MCGILL, OCT 9-10.
So be sure you make your reservations as soon as possible.  If you need
a list of hotels or further information, please let me know.
- Tom Fox
fox@triples.math.mcgill.ca
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Is this result published somewhere?
Date: Wed, 15 Sep 93 16:33:27 +0200
From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens)


Can somebody give me a reference to a proof of the following
result (or some generalisation of it)? (I need this in my research,
and if a proof is published somewhere, I don't have to do the proof
myself)

Let C be a small category and let F:C -> Set be a functor.
Then, the slice category

     Fun(C,Set)/F

is equivalent with

    Fun(G(C,F),Set)

where G is the Grothendieck-construction for Set-valued functors
as described in "Category theory for computing science" (Barr and Wells)
chapter 11.

As a simple example, take C=1, and one becomes the well-known fact that
Set/A is equivalent with Fun(A,Set) (A regarded as a discrete category).

If this is "well-known" among category-theorists, without any published
proof around, please let me know this too.

Thanks in advance,

Frank Piessens
Dept. of Computing Science
Katholieke Universiteit Leuven
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Is this result published somewhere? (3 posts)
Date: Thu, 16 Sep 93 12:25:37 +1000
From: street@macadam.mpce.mq.edu.au

>From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens)
>Can somebody give me a reference to a proof of the following
>result:
>Let C be a small category and let F:C -> Set be a functor.
>Then, the slice category Fun(C,Set)/F is equivalent with
>Fun(G(C,F),Set) where G is the Grothendieck-construction . . .

This result has been known to category theorists for a long
time, and probably appears somewhere in SGA 4. For a published
reference with two-line proof (given a little knowledge of
fibrations) see my Proposition (7.3) page 293 of

"Cosmoi of internal categories" Transactions AMS 258 #2 (1980)
271 - 318.

This result was used to produce an algorithm for finding all internal 
full subcategories of a presheaf category.

Moreover, I pushed up the equivalence to the case where C supports a
sketch (I called sketches "Gabriel theories" in that paper with
a footnote giving the other terminology). If  F is a model of
the sketch, one obtains a sketch on your G(C,F), and an equivalence
                  Mod(C,Set)/F ~~~ Mod(G(C,F),Set).
See Proposition 7.21 on page 297.

I used this result to produce an algorithm for finding all internal 
full subcategories of certain locally presentable categories (eg, Cat).
I also used it to characterize the sketches whose model categories
had cartesian closed slice categories (Theorem 7.25).

[I mention the sketch result because sketches seem to be of interest to
computer scientists.]

An interesting example of the original equivalence is the case where C 
is a group and F is a G-set, so that G(C,F) is the wreath product; or,
if F is a G-module, G(C,F) is the semidirect product.
 
Regards,
Ross

++++++++++++++++++++++
Date: Thu, 16 Sep 1993 12:11:30 +0100 (BST)
From: Roy Crole <rlc@doc.ic.ac.uk>

In reply to Frank Piessen's question about the
equivalence of [C,Set]/F and [G(F),Set]. I
would say that this result is well known,
though I am not so familiar with the original
literature. 

A discrete fibration p : E--->C (over C) is a functor
for which given any morphism f : A---> A' in C and X'
in E with p(X') = A', there is a unique v : X---> X'
in E with p(v) = f.

Then there is a category DFib/C of discrete
fibrations over C, and it is well known that

    DFib/C <--equiv--- [C^op, SET]  :  G ,

this being the analogue of the more commonly cited
result for split fibrations. One half of the above
equivalence is given by the Grothendieck
construction, which induces an equivalence on the
slices

  ( DFib/C ) / ( G(F)---> C) <---equiv--- [C^op,Set] / F.   

If p = p'' o p' is a composition of discrete
fibrations, then p' is a discrete fibration; thus

  ( DFib/C ) / ( G(F)---> C) equiv DFib / G(F)

and this gives the result (modulo op's !)

Of course, proving the result directly is easy
enough, I guess.

Roy Crole

+++++++++++++++++++
From:   Richard Wood <rjwood@cs.dal.ca>
Date:   Thu, 16 Sep 1993 08:39:57 -0300

The result appears in Diaconescu's thesis  (Dalhousie '73) with
`Set' generalized to an elementary topos, Proposition I.1.5 .
From the preceding text one gets the impression that the result
was already well known for `Set' but none of Diaconescu's 
references are likely sources.

Diaconescu's proofs of results about "internal category theory"
contain truly marvelous diagrams the like of which we will
probably not see again until various TeX issues are settled. At
the time it seems that rigor demanded them but that was before
indexed categories, fibrations and languages were well understood.
It is now clear that a careful proof of the result in question,
written as for sets, suffices.

It would be useful to have the result and similar ones proved
in a text or expository paper. I do not know of one.

RJ Wood
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Octoberfest
Date: Sun, 19 Sep 93 17:15:03 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Here is the final list of speakers.  No more will be added.  As you see
we have 19 and that is really too many already.  The talks will begin at
9:00 each morning and the Sunday session will end at 2:40.  A detailed
schedule will be distributed in a few days.
 
Michael
 
Giulio Katis <katis_p@maths.su.oz.au>
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 <>
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>
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>
Tau Categories and Logic Programming
 
 
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!)
 
 
Jim Otto <otto@triples.math.mcgill.ca>
Categories and complexity
 
 
Phil Scott <SCPSG@acadvm1.uottawa.ca>
Coherence and Undecidability for CCC's
 
Abstract:  (Joint Work with M. Okada) We show the equational theory of
simply typed lambda calculus with strong natural numbers object is
undecidable, thus the coherence problem for equality of arrows in the
free ccc with NNO is undecidable.  We study the rewriting theory (made
equational by Lambek's use of Mal'cev operators) and prove in fact the
appropriate lambda calculus is not Church-Rosser, but is Strongly
Normalizing.  The latter proofs require heavy rewriting techniques.
 
 
Jonathon Funk <jfunk@morgan.ucs.mun.ca>
The display locale of a cosheaf
 
 
Peter Freyd <pjf@saul.cis.upenn.edu>
Hardware design and free allegories.
 
Kimmo Rosenthall <ROSENTHK@gar.union.edu>
TBA
 
 
Martin Markl <>
TBA
 
 
Wim Ruitenburg <wimr@mscs.mu.edu>
Yet another constructive logic
 
 
Andre Joyal <joyal@mipsmath.math.uqam.ca>
How to complete a category by freely adjoining all limits and colimits
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Modules list
Note from Moderator:

Apologies to subscribers and the network gods for the typing error which 
resulted in resending the categories and baseball message (the Expos were 
four games back this morning.)

++++++++++++++++++++++++++++++++++++++++++++++++++++++

Date: Sun, 19 Sep 93 13:59:16 -0700
From: rowan@garnet.berkeley.edu

The response to my proposed modules mailing list has been very gratifying
(almost 20 people).  I have decided to go ahead with it, at least for a
trial period.

The focus of the mailing list will be modules (Beck's definition of an 
A-module,
for an object A in a category C, is an abelian group object in the category of
objects of C over A) and the corresponding pointed set objects.  I have my
own equivalent definition, and call the pointed set objects pointed overlaying
algebras.  I am personally most interested in modules and pointed overlaying
algebras over universal algebras A, and interactions with the structure of A.
(For example, congruences of A naturally give rise to pointed overlaying
algebras, and frequently to modules.)  However, the subject where this theory
first got started was cohomology theories.

To start with, there are two activities I want to pursue:

(1) Discussion.  Please contribute opinions, queries, announcements,
whatever, concerning modules and pointed overlaying algebras.

(2) Creating a bibliography of articles on this subject, with brief abstracts.
Please let me know of your work in this area, and how preprints may be 
obtained.
Hopefully, I will be making this bibliography available on the network.

To become a charter subscriber, send e-mail to rowan@garnet.berkeley.edu.
I am not yet ready to receive contributions, and will make another 
announcement
when I am ready.

Thank you,  Bill Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: This attached memo
Date: Thu, 23 Sep 93 08:02:22 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Can someone help this guy out with this?

> Date: Thu, 23 Sep 93 12:45:19 +0200
> From: ldup@alcbel.be (Luc Duponcheel)
> To: barr@Math.McGill.CA
> 
> Michael,
> 
> 
> I have a simple question. I work in the following `framework' :
> 
> 
> A category which is such that all Hom(A,B) are themselves categories 
> and having the following properties :
> 
> 
> first some notation
> -------------------
>  
>   morphisms are denoted as F : A -> B, G : C -> D, ...  
>   and their composition is denoted as GF
> 
>   morphisms in Hom(A,B) (called transformations) 
>   are denoted as alpha : F -> G, beta : H -> K 
>   and their composition is denoted as beta . alpha
> 
> 
> here come the actual properties
> -------------------------------
> 
> 1) for all transformations alpha : F -> G where F : A -> B and G : A -> B, 
>    and all morphisms H : B -> Y there exists a transformation 
>    alpha H : FH -> GH.
> 
> 1a) GF alpha = G (F alpha)
> 1b) F (beta . alpha) = F beta . F alpha
> 
> 2) for all transformations alpha : F -> G where F : A -> B and G : A -> B, 
>    and all morphisms H : X -> A there exists a transformation 
>    H alpha : HF -> HG.
> 
> 2a) alpha GF = (alpha G) F
> 2b) (beta . alpha) F = beta F . alpha F
> 
> 3) G (alpha F) = (G alpha) F
> 
> 
> 
> 
> 
> BTW
> ---
> 
> a transformation beta : H -> K is natural if for all transformations 
> alpha : F -> G one has  K alpha . beta F = beta G . H alpha
> 
> 
> 
> 
> 
> These axioms are a subset of the ones used for 2-categories.
> The transformations do not need to be (but may, of course, be) natural.
> I do not, a priory, need any `horizontal' composition "*" of 
transformations.
> 
> 
> 
> One of the results which I want to prove in this framework
> is just the fact that certain transformations (who do not a priory need to 
> be natural at all) are nevertheless, under certain conditions (of a 
> different nature) natural.
> 
> 
> Is there any *name* for this `framework'?
> 
> 
> I could call it *categories with transformations* but if there is
> any other name which is commonly used, then I would appreciate if you
> can inform me about it.
> 
> 
> 
> Thanks!
> 
>    
> 
> Luc.
> 
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Memo of Luc Duponcheel
Date: Fri, 24 Sep 93 11:04:09 +1000
From: street@macadam.mpce.mq.edu.au

Luc seems to be describing what, in our work on rewrites etc, Eilenberg 
wanted to call a one-and-a-half-category, and I changed (and I'm not
sure how Sammy feels about it) to sesquicategory when writing the article
"Categorical structures" for "Handbook of Algebra" Volume 2 
(Elsevier, North Holland). [The preprint is dated Nov 1992 but the
volume probably won't appear until late 1994!]

There are two symmetric monoidal closed structures on the category
Cat of categories. One is the usual cartesian closed structure where
the internal hom [A,B] is the usual category of functors A --> B
and NATURAL transformations between these. The other closed structure
(which I call the "funny" one) has internal hom [[A,B]] the category
of functors and transformations between these (which consist of the
data for a nat trans without the naturality requirement). This funny
structure WAS an example in Eil-Kelly "Closed categories" La Jolla
1965. Carolyn Brown has used the funny structure in her work on
Petri nets. 

Sesquicategories are V-categories where V is Cat with the funny monoidal
structure. John G. Stell <john@cs.keele.ac.uk> tells me he has used 
sesquicategories in computer science, and apparently independently
came up with the same name for them.

Perhaps of future interest to those interested in sesquicats are the
various monoidal structures on 2-Cat. For example, there is John Gray's
tensor product of 2-categories, where "natural", instead of being
dropped, is replaced by "lax natural". A recent paper of Gordon, Power,
Street looks at the case where V = 2-Cat with the monoidal structure
obtained by replacing "natural" by "pseudonatural"; we prove that every 
tricategory is equivalent (in the approp sense) to a V-category. 

Regards,
Ross 

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: tentative timetable for Octoberfest
Date: Fri, 24 Sep 93 07:27:44 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Saturday, October 9
----------------------------------------------------------------------------
Chair: J. Lambek
 9:00 -  9:45   Andre Joyal, How to complete a category by freely
                adjoining all limits and colimits
 9:50 - 10:20   Martin Markl, Deformations of everything
 
10:20 - 10:50   Break
 
Chair: A. Joyal
10:50 - 11:20   Wim Ruitenburg, Yet another constructive logic
11:25 - 11:55   L Gaunce Lewis Jr, Equivariant Freudenthal suspension
                theorem
12:00 - 12:30   Robin Cockett, Copy Categories
 
12:30 -  2:00   Lunch
 
Chair: R. Wood
 2:00 -  2:30   Richard Wood, Distributive adjoint strings
 2:35 -  3:05   Phil Scott (and M. Okada), Coherence and Undecidability
                for CCC's
 3:10 -  3:40   Rick Blute, (Robert Seely and Robin Cockett)
                Contextual Logic
 
 3:40 -  4:10   Break
 
Chair: R. Rosebrugh
 4:10 -  4:40   Andreas Blass, TBA
 4:45 -  5:15   Bob Gordon (and John Power), Enrichment Through Variation
 5:20 -  5:50   Jon Beck, TBA
 
 6:30 -         Reception
 
 
 
Sunday, October 10
---------------------------------------------------------------------------
Chair: T. Fox
 9:00 -  9:45   Till Plewe, When a locale product of metrizable spaces
                is spatial
 9:50 - 10:20   Jonathan Smith (and  A. Romanowska), Duality for
                semilattice representations
 
10:20 - 10:50   Break
 
Chair: P. Scott
10:50 - 11:20   Peter Freyd, Hardware design and free allegories
11:25 - 11:55   Stacy Finkelstein, Tau Categories and Logic Programming
12:00 - 12:30   Kimmo Rosenthall, TBA
 
12:30 -  1:00   Break
 
Chair: R. A. G. Seely
 1:00 -  1:30   Jonathon Funk, The display locale of a cosheaf
 1:35 -  2:05   Djordje Cubric, Interpolation property for bicartesian
                closed categories
 2:10 -  2:40   Jim Otto, Categories and complexity
 
 
 
Jon Beck
TBA
 
 
Andreas Blass <ablass@math.lsa.umich.edu>
TBA
 
 
Rick Blute <RBLUTE@acadvm1.uottawa.ca>
Contextual Logic (joint with Robert Seely and Robin Cockett)
 
 
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!)
 
 
Djordje Cubric <cubric@triples.math.mcgill.ca>
Interpolation property for bicartesian closed categories
 
 
Stacy Finkelstein <stacy@saul.cis.upenn.edu>
Tau Categories and Logic Programming
 
 
Peter Freyd <pjf@saul.cis.upenn.edu>
Hardware design and free allegories
 
 
Jonathon Funk <jfunk@morgan.ucs.mun.ca>
The display locale of a cosheaf
 
 
Bob Gordon <gordon@euclid.math.temple.edu>
Enrichment Through Variation (joint with John Power)
 
 
Andre Joyal <joyal@mipsmath.math.uqam.ca>
How to complete a category by freely adjoining all limits and colimits
 
 
L Gaunce Lewis Jr <gaunce@ichthus.syr.edu>
Equivariant Freudenthal suspension theorem
  One of those nice situations when just a little touch of category
theory cleans up a mess in topology.
 
 
Martin Markl <>
Deformations of everything
 
 
Jim Otto <otto@triples.math.mcgill.ca>
Categories and complexity
 
 
Till Plewe <>
When a locale product of metrizable spaces is spatial
 
 
Kimmo Rosenthall <ROSENTHK@gar.union.edu>
TBA
 
 
Wim Ruitenburg <wimr@mscs.mu.edu>
Yet another constructive logic
 
 
Phil Scott <SCPSG@acadvm1.uottawa.ca>
Coherence and Undecidability for CCC's (joint with M. Okada)
  We show the equational theory of simply typed lambda calculus with
strong natural numbers object is undecidable, thus the coherence problem
for equality of arrows in the free ccc with NNO is undecidable.  We
study the rewriting theory (made equational by Lambek's use of Mal'cev
operators) and prove in fact the appropriate lambda calculus is not
Church-Rosser, but is Strongly Normalizing.  The latter proofs require
heavy rewriting techniques.
 
 
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.
 
 
Richard Wood <rjwood@cs.dal.ca>
Distributive adjoint strings
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: change in chairs
Date: Sat, 25 Sep 93 09:37:24 EDT
From: barr@triples.Math.McGill.CA (Michael Barr)

Please make the following changes in chair assignments:
Wood --> Sunday at 9
Fox --> Saturday at 2
Sorry about that.  --MB
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: What do you call this?
Date: Fri, 24 Sep 93 23:51:34 -0700
From: rowan@garnet.berkeley.edu

I call a functor F, such that F is one-one and onto on objects, and
F sends each hom(a,b) ONTO hom(Fa,Fb), a _cofaithful_ functor.  I call it
that because every functor has an essentially unique decomposition as a
faithful functor, followed by a cofaithful one.

The question is, what do YOU call this?  I would like to use the standard
terminology if there is one.

Thanks, Bill Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: On UA-CAT Workshop at MSRI
Note from moderator:

The following was received as a typescript from Saunders Mac Lane, with a
request to post it, so typographical errors are mine. He mentions that his
e-mail does not work well, and I will forward a copy of any discussion which
ensues here to him, but of course he can also be contacted directly by mail.

Bob Rosebrugh
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

From: Saunders@math.uchicago.edu
Date: Sept. 25, 1993

RANDOM THOUGHTS ON THE RECENT UA-CAT WORKSHOP AT MSRI
Saunders Mac Lane   Dept. of Math, Univ. of Chicago, Aug, 1993

In addition to drafting a final report on this workshop, I'll venture here to
ask some provocative questions about the interrelation between Universal
Algebra (UA), Category Theory (CAT) and the connections of both subjects to 
all
(!) the rest of mathematics.  I hope thereby to provoke both strong dissent 
and
some progress and innovation.

First those minor remarks about terminology.

CLONE vs. ALGEBRAIC THEORY. The notion of a clone - for each n, all n-ary
operations of the intended type of algebra - is due to Philip Hall (apparently
unpublished by him). Lawvere's notion of an "algebraic theory" (his thesis,
1963) is a category, objects the natural numbers, maps n-->m given by all
m-tuples of term operations. These two notions are essentially equivalent; the
word "clone" is short and convenient, while the Lawvere notion includes more
composites and so is more flexible and easier to formulate and to generalize. 
It follows one of the tacit slogans of CAT: With every map, present at once
both the domain and the codomain, by this effective presentation making the
whole situation more visible and showing what composites are possible.  This 
is
the useful adage "Draw Diagrams! This art should be easy to learn; but as for
the choice of words, I suppose that both "clone" and "Algebraic theory" will
continue in use, hopefully with a more general recognition of their
equivalence.

"GROUPOID" as the name of a category with every arrow an isomorphism,
apparently appeared first in 1926 in a study of the ideals of algebraic
integers in a non-commutative algebra (Brandt). This notion has vital uses in
many parts of mathematics, as for example for the fundamental groupoid of a
space, and so combining the fundamental groups at each base point.  It appears
also in differential geometry and in mechanics, as a means of transporting
structures along one of the paths of the space at hand.  In UA, on the other
hand, a groupoid has meant a set with a single binary operation--and no
required identities; it may be that this usage first appeared in a book by R.
H. Bruck.  Such objects do need a name, but clearly the categorical use of
"groupoid" is the widest and most important.  I hope that the other use can be
replaced by some other suitable term; possibly "binary algebra" or perhaps 
even
"binoid".

The "empty algebra" has been proscribed in UA. Since there is now occasion to
consider models of algebras not just in sets (possibly in topoi or in regular
categories) this early proscription seems too narrow.


WHAT IS AN ALGEBRA? CAN WE LOOK BEYOND THE PRESENT? The Clone-algebraic theory
notion is now evidently too narrow; no infinitary operations.  The algebras 
for
a Monoid (=triple, as in Linton's lecture) are somewhat more general and much
neater (adjoints again!, cf. my "Categories Work:"). The algebras described by
sketches are considerably more general.  Recall the lecture by Wells, where a
sketch together with its models is described by a graph, together with a set 
of
diagrams there (to commute in the models), a set of cones (to be limits) and a
set of cocones (colimits). This notion is flexible, especially because it
comprises colimits as well as limits.  The lecture by John Gray indicated how
this works for computer science; see also recent books e.g., by Barr-Wells. 
The
Workshop did not get to consider how standard concepts of UA would work out 
for
sketches.  Much more might develop here!

WHERE DO THE MODELS OF ALGEBRAS LIE? Classical model theory says: "in sets,"
with a spectrum.  There have been a few starts at taking models in a general
topos.  Does this provide for greater "Variety"?
 
MALTSEV ALGEBRAS are conventionally described by those familiar term
identities--but their bearing on the composition of relations is relevant,
while a recent paper of Carboni, Lambek and Pedicchio (JPAA, 69, 1991)
indicates how a category of such algebras can be characterized as a kind of
regular category.  This offers a striking connection; see also a preprint,
Kelly et al, on Goursat categories.  Many prospects might open up here. 

TAME CONGRUENCE THEORY provides remarkable and surprising results for finite
algebras (those astonishing five types of possible quotients for a covering). 
But finite algebras have a somewhat limited scope; perhaps something equally
"Tame" could be managed for some classes of infinite algebras, possibly by the
well-tried method of introducing a topology?

FINITE CLONES on two elements (E. L. Post) bear on the problems of "truth
values"; but there appear to be too many clones on bigger finite sets; it is
not yet clear to me how they relate to other mathematics.

COMMUTATOR THEORY is a fascinating development, especially in view of its
relation to commutators in group theory--and what about Lie algebras and the
Lie bracket here? At the end of the workshop M. C. Pedicchio mentioned to me a
somewhat more categorical description of such commutator theory.

SUBOBJECT LATTICES and CONGRUENCE LATTICES seem to arise almost separately in
UA.  One merit of categories is that they bring together the inclusions
(subobjects) and the epimorphisms (quotients), both appearing as arrows.  Does
this view have consequences? Is it simpler?

INTERACTIONS.  Some of the most striking developments of mathematics come from
unexpected connections (collisions) between different subjects.  I know of
several such collisions in CAT: The use of categories in the drastic
reformulation of algebraic geometry by Grothendieck, the connection to Quillen
Model structures in topology (Tierney's lecture) and the recent use of
coherence theorems for commuting diagrams as they appear in quantum field
theory and quantum groups (which are really not groups but deformed Hopf
algebras).  Then there is computer science: sketches, categories expressing
data types and the newer uses of topoi there.  I do not know of similar 
outside
connections for UA. How should one search and in which other fields? In Logic?

SIMPLICIAL SETS were designed for calculations for Eilenberg-Mac Lane spaces,
defined as functors, and now are used as surrogate spaces!

CONCEPTS: Emmy Noether's "modern algebra" achieved much, and in particular
emphasized the astute use of general concepts--the homomorphism theorems, for
example, to illuminate and make understandable known mathematical results. 
This was in particular the case with Galois theory, which before had been
expressed in terms of obscure "substitutions", but was then organized in terms
of Automorphisms to become more perspicuous, as later in Emil Artin's
presentation.  The same clarification came to ideal theory, done jointly for
algebraic numbers and for algebraic varieties.  I personally hold that a major
role of "abstract" mathematics is this sort of clarification and
understanding--both for its own sake and for the new results it can and does
encourage.  The recent reformulation of algebraic geometry is a striking
example of this.

The notion of an ADJOINT FUNCTOR (for posets, Galois correspondence) is one of
the most pervasive and effective notions of CAT (recall Joyal's lecture on the
Witt vectors as an adjoint).

TWO-DIMENSIONAL categories (with 2-cells to represent things such as 
homotopies)
and the related higher dimensional categories, originally seemingly
complicated, have now found much use (Street's lecture), as in the solutions 
of
Yang-Baxter equations, and in coherence theorems.

PROBLEMS vs. CONCEPTS. Sometimes mathematics is viewed as the construction and
astute solution of hard problems.  When they are indeed of central interest 
and
make use of powerful methods, as in the recent case of Fermat, this is
splendid.  On the other hand, not every hard problem is necessarily relevant 
to
the progress of knowledge.  How to select?

The PARTICULAR vs. the GENERAL.  Some categorical studies are troubled by a
sort of "dual" objective--the study of special problems vs. the formulation of
general notions directed at wide aspects of mathematics - what is a theory, 
for
example.  Lawvere's lecture at the Workshop emphasized the general, in the
guise of a Doctrine (a category with structure) or a hyperdoctrine.  These
questions may have some bearing on education: in teaching, should we present
just the particulars and just the corresponding skills, or somehow try to
convey the general in its particular guise.  Such questions are provocative 
and
contentious, but nevertheless useful. Learn it all young! This is now 
possible;
recently, Lawvere and Schanuel have prepared an elementary text on "Conceptual
mathematics." 

A TOPOS is a category with good structure (finite limits, exponentials.
subobject classifier).  It encompasses sheaf theory, yields an internal logic
with a Heyting algebra of truth values, while sheafification gives a
formulation of Cohen forcing, Topoi have recently been used in modelling
computer science; a topos thus exemplifies many connections.

Thus arises my hope that we can search here and elsewhere for decisive new
interactions.  The growth of ideas can be splendid!

Saunders Mac Lane
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: one minor addition
Date: Tue, 28 Sep 93 21:02:08 EDT
From: James Stasheff <jds@tramp.math.unc.edu>

to Saunder's description of clones,etc.
recent work in String Field Theory has found use not only for operads
(the pieces which assemble into a monoid) corresponding to n-ary ops
but something that is new (at least to me):
think of rooted trees as desribing n-ary ops with composition given by 
grafting
root to branch e.g. associative algebras can be described by planar trees
and Lie algebras by abstract trees
        then trees with edge lengths on the internal edges can be used to
described strong homotopy analogs
                now what if we allow more general graphs with grafting
allowed between any two branches as well
        has anyone other than a string field theorist ever seen such algebra??
                jim
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Announcement
Date:         Wed, 29 Sep 93 11:58:42 EDT
From:         SCPSG@acadvm1.uottawa.ca

Dear Colleagues:

The annual winter meeting of the Canadian Mathematical Society
is being held this year in Ottawa, Ontario,  December  11- 13. I was
asked  to organize one of the special sessions, entitled
Categorical Logic and Theoretical Computer Science.  The
speakers and titles are:

M. Barr:  Pontrjagin Duality Revisited
A. Blass: Geometric Morphisms
R. Cockett: Categorical Recursion Theory
P. Freyd:  tba
A. Joyal: Completion of Categories and Communication Games
M. Makkai: Contributions to the Theory of Doctrines.
R. Pare:  Dinatural Numbers.
A. Pitts (Plenary Speaker): Category Theory and the Semantics of
                      Programming Languages.
R. Rosebrugh: Functorial Aspects of Relational Databases.
A. Scedrov:  Relators
R. Seely: Coherence of Bimonoidal Categories


In case anyone is thinking of coming, I enclose some hotel data.
The fee schedule, other technical information about the meeting,
other special sessions, etc. are available in the CMS Notes (September
issue), or you can contact me. Fees are cheaper if you register soon.

                   Philip J. Scott
                   Dept. of Mathematics
                   University of Ottawa
                   585 King Edward
                   Ottawa, Ont. Canada  K1N 6N5

                   e-mail:  scpsg@acadvm1.uottawa.ca
                   FAX:    613-564-3822


---------------------------------------------------------

THE MEETING

The meeting is being held in the Westin Hotel, a somewhat upscale
hotel in downtown Ottawa, connected to Rideau Centre Mall.  There
are lots of shops and restaurants in the mall and no need to
wear heavy clothes if you stay indoors.  Rideau Mall is right
downtown (but, for walkers,  December weather can be
quite unpredictable). For American visitors: all prices quoted are
CANADIAN DOLLARS, currently 75 cents U.S.

Hotels:

Westin Hotel (1-800-228-3000):  Conference Centre.
Winter Canadian Math Society (CMS) prices:
 $99.00 Can. (single), $109.00 Can. (double)  per night.
Make sure you get the CMS prices.

The following hotels/BB are very close by (note the
Novotel and B & B below):

NOVOTEL (1-800-221-4524):  across the street
from the Rideau Centre: 2 minutes walk to the Westin.
Not as fancy as the Westin, but modern.
I got a SPECIAL PRICE for rooms: (must be booked
before Nov. 10: Ask for Canadian Math. Society rate:
Jim Tanguay, Manager)   $75.00 Can. (single or double) per night.

LES SUITES (1-800-267-1989):  Executive Apartment Suites.
These have full apartments, and are brand new.
Across the street from Rideau Centre, next door to
Novotel.   Weekend Specials:
1 bedroom apartment:  $85.00 Can. per night. 2 bedroom
apartment:   $125.00 Can. per night.

B&B:  Gasthouse Switzerland:  (613-237-0335)
A European-run Bed and Breakfast.  About 3 minute walk to Rideau
Centre, less than 5 minutes to the Westin hotel.   $58.00 single
room, with breakfast and private bath. $68.00 double room, with
breakfast and private bath.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Memo of Luc Duponcheel
Date: Wed, 29 Sep 93 08:58:15 +0100
From: grandis@cartesio.dima.unige.it. (Marco Grandis)

   I used recently the following hierarchy of notions (one of them
coinciding with L.D.'s one) as
abstract settings for homotopical algebra. A synthesis is given in Cahiers
Top. Geom. Diff. Cat.  33 (1992, 135-175 (ch. 5-6); a longer 1991 preprint
on the subject is no longer available, but a new version will soon be
ready.


- "h-category", or "category with homotopies".  (This notion goes back to
K.H. Kamps (Manuscripta Math. 3 (1970), 237-255), who used it in a slightly
different, equivalent form under the name of "generalized homotopy system";
it is an extension of a Kan "homotopy system", or category with cylinder
endofunctor)

   Formally, an h-category is a category enriched over Reflexive Graphs,
with a suitable monoidal closed structure.
   Concretely, an h-category has objects, maps  f: A -> B  and "cells" (or
homotopies, or transformations)  alpha: f -> g (for f: A -> B).  Objects
and maps form a category; further, there is a "vertical identity"  id f: f
-> f  for every map and a "reduced horizontal composition" of cells with
maps:

                      k alpha h            (for  h: A' -> A,  k: B -> B')

under the obvious axioms for identities and associativity

   (id B) alpha (id A)  =  alpha,
   k (id f) h  =  id (kfh),
   (k'k) alpha (hh')  =  k' (k alpha h) h'.

   Of course one may separate the left and right composition of cells with
maps: k alpha, alpha h. Note that there is no vertical composition.


-   h1-category  =  h-category + vertical involution (under some weak axioms)


-   h2-category  =  h-category + vertical composition (under some weak
axioms; for instance the vertical comosition is not required to be
associative)

   (strict h2-category  =  h2-category with axioms for vertical identities
and vertical associativity
                                  =  sesquicategory in the sense of R.
Street's reply
                                  =  (probably) "category with
transformations" in the sense of L.D.'s message

-   h3-category  =  h-category + vertical involution and composition (under
some weak axioms)

   (strict h3-category  =  h3-category with axioms for vertical identities,
vertical inverses and
                                      vertical associativity
                                  =  category enriched over groupoids, with
the "funny" structure
                                  =  sesquigroupoid?


-   h4-category  =  h3-category + second-order homotopy relation making it
a sort of relaxed 2-category.


  To write down the complete definitions would be too long here. But the
following examples should be sufficient to make them clear, and also to
motivate them

a) Top  =  Topological spaces, continuous maps and homotopies. It is h4,
not strict - the vertical identities, involution and composition just "work
well" up to second-order homotopy.

b)  C*A  =  Chain complexes (over a preadditive category A), chain maps,
homotopies. It is h4 and strictly h3 (the vertical composition of
homotopies is obtained from the sum, and works well in the strict sense).
   This sesquicategory could be of interest for L.D., as an algebraic model
of his situation. The presence of a vertical involution (strictly well
behaved) should also be of interest.

c)   Dga  =  Differential graded algebras (over some unital ring), their
homomorphisms and homotopies. It is just an h-category - the
multiplicativity conditions on homotopies prevent to reverse or add them.


     Best regards,        Marco Grandis




++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: AMAST Workshop on Real-Time Systems
Date: Wed, 29 Sep 93 17:14:59 BST
From: Joseph.Goguen@prg.oxford.ac.uk

The following seems relevant to Mac Lane's remark that he does not know of
applications of universal algebra.  In fact, in Computer Science, the line
between UA and CATH is not so firmly drawn as in Mathematics, and there is
actually a bias towards using the least sophisticated formalism possible for
a given application.

Joseph
&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Signature File &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Joseph A. Goguen, Professor of Computing Science, Programming Research Group,
University of Oxford, 11 Keble Road, Oxford OX1 3QD, United Kingdom.

email: Joseph.Goguen@prg.ox.ac.uk [internet] -- usually also works in the
  UK, but if not, try Joseph.Goguen@uk.ac.ox.prg

phone: 272567 [my office]; 272568 [secy]; 273838 [PRG office]; 273839
  or 272582 [FAX].  From USA, dial 011-44-865-...; from UK, dial (0865)-...

******************************************************************************
******************************************************************************
Date: Tue, 28 Sep 1993 17:57:23 +0100
To: concurrency@nl.cwi
Subject: AMAST Workshop on Real-Time Systems
From: Teodor Rus <rus@edu.uiowa.cs.herky>
Sender: fritsv@nl.cwi


\documentstyle [11pt]{article}
\textwidth = 6.5in
\textheight = 8.5in
\topmargin = -0.3in
\topskip = 0in
\oddsidemargin = -0.1in
\evensidemargin = -0.1in
\addtolength{\parskip}{0.5ex}
\begin{document}
\begin{center} {\LARGE {\bf PROGRAM}}\\
\ \ \\
{\Large {\bf First AMAST International Workshop on Real-Time 
Systems}}\footnote
{This conference is sponsored by grants from the National Science Foundation, 
Office of Naval Research,
ESPRIT Basic Research Programme, University of Iowa, and University of Twente.
}\\
{\Large {\it 1--3 November 1993, Iowa City, Iowa, USA}}
\end{center}

\medskip\noindent{\bf Organizing Committee:}
\begin{quote}
\begin{tabbing}
Maurice Nivat, University of Paris VII, France\\
Charles Rattray, University of Stirling, Scotland\\
Teodor Rus, University of Iowa, Iowa City, IA, USA\\
Giuseppe Scollo, University of Twente, The Netherlands \\
\end{tabbing}
\end{quote}

\noindent {\bf Aim:}

Dedicated real-time applications form one of the areas of great 
practical accomplishment of current computer technology. 
Real-time applications, however, bring to the fore new and intriguing 
questions
regarding program specification, verification, and development. 
Correctness of solutions to the problems raised by real-time programming
is particularly important due to the catastrophic nature
of failure in real-time systems. This motivates  
the extensive work in the past decade on the formal theory of
specification, verification, and development of real-time systems.
At the same time, the AMAST movement, initiated in 1989 and aiming to 
use algebraic methodology for the development of software
technology, has started to show practical results. The goal of this
workshop is to expand the AMAST results
to real-time system development, by:  
\begin{enumerate}
\item
Providing a forum for a dialog on the suitability
of using algebraic methodology for real-time system development. 
\item
Tracing the directions of a unifying approach for real-time system 
development within the framework provided by universal algebra. 
\item
Promoting the integration of real-time system development
within software technology based on the new
algebraic methodology which is emerging from an AMAST approach. 
\end{enumerate}
It is the intention of the organizers to publish the
research reported at this workshop in a 
{\it Handbook on Real-Time System Development} in the AMAST
Series in Computing. The feasibility of this project will be discussed
in the special sessions scheduled during the workshop. We invite
contributions to these discussions and submissions to the
handbook from all attendees of the workshop. 

\medskip\noindent All meetings of this workshop will take
%place in the room 345 (Northwestern) 
place at the Iowa Memorial Union. 
Each talk presented at this workshop
will be 50 minutes long, followed by 10 minutes discussion.
Supplementary discussion time will be provided in special sessions.

\medskip\noindent 
Continental breakfast will be served each morning 8:30--9:00 at the
meeting room. Lunch will be served each day 12:30--1:30 in BF 236, 
Second Floor, Iowa Memorial Union.
\newpage
{\small
\noindent {\bf Monday, November 1-st, 9:00--12:30 Session 1}

\medskip\noindent
$\spadesuit$ 8:00--8:30 Registration and breakfast

\medskip\noindent$\spadesuit$ 8:30-9:00 Opening address by
Prof. David J. Skorton, Vice President for Research, The 
University of Iowa.

\medskip\noindent 1. 9:00--10:00 
{\it Finite Automata, Omega-Languages and Distributed Systems}
by Maurice Nivat, Universit\'{e} Paris 7, France.

\medskip\noindent 10:00--10:15 Coffee break 

\medskip\noindent 2. 10:15--11:15
{\it Issues in the Specification and Verification of Telephone Systems}
by Luigi Logrippo, Department of Computer Science, 
University of Ottawa, Ottawa, Ont, Canada K1N 6N5.  

\medskip\noindent 11:15--11:30 Coffee break

\medskip\noindent 3. 11:30--12:30 
{\it On the Design of Timed Systems}
by Juan Quemada, Departmento de Ingineria Telematica,
Universidad Politechinica de Madrid, Spain.

\medskip\noindent 12:30--1:30 Lunch break
%\newpage

\medskip\noindent {\bf Monday, November 1-st, 1:30--5:00 Session 2}

\medskip\noindent 4. 1:30--2:30
{\it Visual Tools for Verifying Real-Time Systems}
by Jonathan Ostroff,  
Department of Computer Science, York University,   
4700 Keele Street, North York,  Ontario, Canada, M3J 1P3.         

\medskip\noindent 2:30-2:45 Coffee break

\medskip\noindent 5. 2:45--3:45
{\it Integrating State Machines, Temporal Logic, 
and Algebraic Models of Data}
by Armen Gabrielian, UniView Systems, Mountain View, California, USA.

\medskip\noindent 3:45-4:00 Coffee break

\medskip\noindent 6. 4:00--5:00 
{\it Towards Full Timed LOTOS}
by Tommaso Bolognesi, C.N.R. Istituto CNUCE, 
36, Via S. Maria, 56100 - Pisa, Italy.

\medskip\noindent 5:00--8:00 Dinner

\medskip\noindent$\bullet$ 8:00--10:00 Special session
%\newpage 

\medskip\noindent {\bf Tuesday, November 2-nd, 9:00--12:30 Session 3}

\medskip\noindent 7. 9:00--10:00
{\it Refining and Abstracting Time Information}
by Steve Schneider, Oxford University, England. 

\medskip\noindent 10--10:15 Coffee break

\medskip\noindent 8. 10:15--11:15
{\it  Real-Time System = Discrete System + Clock Variables}, Part I
by Rajeev Alur, AT\&T Bell Labs, Murray Hill, New Jersey, USA and 
Tom Henzinger, Department of Computer Science, Cornell University,
Ithaca, New York, USA.

\medskip\noindent 11:15--11:30 Coffee break

\medskip\noindent 9. 11:30--12:30 
{\it  Real-Time System = Discrete System + Clock Variables}, Part II
by Rajeev Alur, AT\&T Bell Labs, Murray Hill, New Jersey, USA and 
Tom Henzinger, Department of Computer Science, Cornell University,
Ithaca, New York, USA.

\medskip\noindent 12:30--1:30 Lunch
%\newpage

\medskip\noindent {\bf Tuesday, November 2-nd, 1:30--5:00 Session 4}

\medskip\noindent 10. 1:30--2:30 
{\it An Experience with the Formal Description in LOTOS and 
Prototyping of the Airbus A320 Flight Warning Computer}
by Hubert Garavel, VERIMAG, Miniparc-ZIRST, rue Lavoisier, 
38330 Montbonnot St Martin, France and
Rene-Pierre Hautbois, Aerospatiale A/DL/EP, M 8621, 
316 route de Bayonne, 31060 Toulouse cedex 03 France. 

\medskip\noindent 2:30--2:45 Coffee break

\medskip\noindent 11. 2:45--3:45 
{\it Specification and Proof in Real-time CSP}
by Jim Davies, Department of Computer Science, University of Reading,
Reading RG6 2AH, England.

\medskip\noindent 3:45--4:00 Coffee break

\medskip\noindent 12. 4:00--5:00
{\it The Priority Inversion Problem and Real-Time Symbolic 
Model Checking}
by Edmund Clarke and Sergio V. Campos, Department of Computer Science,
Carnegie Mellon University, Pittsburgh, PA, USA.

\medskip\noindent 5:00--8:00 Dinner

\medskip\noindent 8:00--10:00 Special Session
%\newpage

\medskip\noindent {\bf Wednesday, November 3-rd, 9:00--12:30 Session 5}

\medskip\noindent 13. 9:00--10:00,
{\it Using Synchronized Transition Systems to 
Develop Real-Time Software: An Experiment}
by Didier Begay, Universit\'e Bordeaux I, LaBRI, 351, cours de la 
Lib\'eration 33405 Talence, France.

\medskip\noindent 10:00--10:15 Coffee break

\medskip\noindent 14. 10:15--11:15 
{\it Verification of the Easylink Protocol}
by Frits Vaandrager and Indra Polak, CWI and University of Amsterdam,
The Netherlands.

\medskip\noindent 11:15--11:30 Coffee break

\medskip\noindent 15. 11:30--12:30 
{\it Performance Analysis and True Concurrency Semantics}
by Ed Brinksma, Joost-Pieter Katoen, Rom Langerak, and Diego Latella,
Department of Computer Science, University of Twente, The Netherlands.

\medskip\noindent 12:30--1:30 Lunch
%\newpage

\medskip\noindent {\bf Wednesday, November 3-rd, 1:30--5:00 Session 6}

\medskip\noindent 16. 1:30-2:30 
{\it Using Iterative Symbolic Approximation for Timing Verification}
by David Dill and Howard Wong-Toi, 
Department of Computer Science, Stanford University, Stanford, CA, USA.

\medskip\noindent 2:30--2:45 Coffee break

\medskip\noindent 17. 2:45--3:45 
{\it Analysis, Synthesis, and Optimization of Real-Time Systems
in a Temporal Logic Framework}
by Dan Ionescu, Department of Electrical Engineering,
University of Ottawa, Ottawa, Ontario, Canada K1N 6N5.

\medskip\noindent 3:45--5:00
Administrative matters and departure 
}
\newpage
\begin{center}
{\Large {\bf General Information}}
\end{center}

\medskip\noindent All speakers at this workshop have been invited.
Their presentations represent some of the best known research directions in 
real-time system
development and we hope that their work will be of interest to a large 
audience. 
So, we would encourage all those who believe that they can benefit from these 
presentations to attend this workshop, to contribute 
to the discussions, and to further the development of real time systems.
% by discussions during these three days of presentations.

\medskip\noindent {\bf Location:}
The conference will be held at the Conference Center of
the University of Iowa\footnote{The University of Iowa does not 
discriminate in its educational programs and activities on the basis
of race, national origin, color, sex, age, or disability. The University
also affirms its commitment to providing equal opportunities and equal
access to University facilities without reference to affectional preference.
For additional information on nondiscrimination policies, contact
the Coordinator of Title IX and Section 504 in the Office of Affirmative
Action, telephone (319)335-0705, 202 Jessup Hall, The University of
Iowa, Iowa City, Iowa 52242-1316. If you are a person with disability
who requires reasonable accommodations in order to participate
in this program, please contact the sponsoring department at (319)335-3231
to discuss your needs.}, Iowa Memorial Union. All meetings
will be held in Room 345, Northwestern, at that location. 

\medskip
\noindent {\bf Transportation}

\begin{enumerate}
\item
The airport that services Iowa City is at Cedar Rapids,
25 miles from Iowa City. 
The closest international
airport from Cedar Rapids is Chicago. 
Limousine services between Cedar Rapids airport
and Iowa City are available.
\item
Interstate 80 is the easiest access route to Iowa City. 
Exit 244, Dubuque Street, leads you to downtown Iowa City.
\end{enumerate}

\noindent {\bf Climate:}
It usually rains in Iowa City on November 1-st. However,
considering the amount of rain we have had so far maybe it will
be sunny this time.

\medskip\noindent
{\bf Registration fees}: \$150; this includes breakfast, 
lunch, coffee and refreshments, and the program and other documents
distributed at the conference site. 

\medskip\noindent {\bf Hotel Reservation:}
For hotel reservation please call  319-335-3513, Iowa House,
indicating that you are attending the First AMAST International
Workshop on Real-Time Systems.
A block of rooms have been already reserved for you
at \$52-single and \$58 double, a night.
They will be assigned to the attendees on the basis of first
come first served. The alternative is Holiday Inn -- downtown Iowa
City -- which is within walking distance from the Iowa Memorial Union. The
number to call is 319-337-4058, reservations.   
The Center for Conferences and Institutes
is handling the registration and the other arrangements. 
For more information about reservation and registration contact: 
 \begin{tabbing}
Bobby C Davis  or Lisa Barnes\\                       
Center for Conferences and Institutes \\
The University of Iowa, Iowa Memorial Union \\
Iowa City, Iowa 52242                 \\
Phone (319)335-3220                   
\end{tabbing}
\end{document}



