Subject: Re: Name for this concept?
Date: Mon, 1 Nov 93 15:58:46 +1100
From: kelly_m@maths.su.oz.au (Max Kelly)

 Bill Rowan asks:
"What do you call it if you have a category C, and you have a class X of
arrows of C such that if x in X, then gxf in X for all composable isomorphisms
f and g.  A functor C->C' which is one leg of an equivalence takes such a set 
to
another one X' in C', and any functor which is the other leg of the
equivalence takes X' back to X."   
     
  Well, I have called it an IDEAL; see   

G.M.Kelly, On the radical of a category,  Jour. Austral. Math. Soc. 4 (1964), 
299-307

and

G.M.Kelly and F.W.Lawvere,  On the complete lattice of essential
localizations, Bull. Soc. Math. Belgique 41 (1989), 289-319.

Max Kelly.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Name for this concept?
Date: Mon, 1 Nov 1993 12:02:07 +0000
From: sjv@doc.ic.ac.uk (Steven Vickers)

>What do you call it if you have a category C, and you have a class X of
>arrows of C such that if x in X, then gxf in X for all composable 
isomorphisms
>f and g?

I haven't seen a name, but can I suggest calling X a "2-sided sieve"? (Or
"2-sided crible"?) Conceivably, people who know what a sieve is could
hazzard a guess at what "2-sided sieve" means.

Steve Vickers.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Name for this concept?
Date: Sun, 31 Oct 93 14:05:08 PST
From: pratt@CS.Stanford.EDU


        Date: Thu, 28 Oct 1993 21:30:45 -0700
        From: "William H. Rowan" <rowan@crl.com>

        What do you call it if you have a category C, and you have a
        class X of arrows of C such that if x in X, then gxf in X for
        all composable isomorphisms f and g.

This rolls connectedness and abstractness into the one condition, as
follows.  It had therefore better have some independently justifiable
redeeming social value before burdening math dictionaries with its own
name.

1. Dropping "iso" from your condition strengthens it to the notion of
*connected component*.

2. Requiring that X be closed under all automorphisms F:C->C, in the
sense that x in X implies F(x) in X, strengthens your condition to what
might reasonably be called an *abstract class*.  A category with two
distinct isomorphic connected components (e.g. *-->*  *-->*) witnesses
the strictness of this strengthening, in that a single component does
not form an abstract class but does satisfy your condition.

Therefore, as a strong common weakening of "connected component" and
"abstract class" (but not the strongest, being strictly weaker than
their disjunction), it would seem that your condition deserves nothing
shorter than "connected-abstract class."

        A functor C->C' which is
        one leg of an equivalence takes such a set to another one X' in
        C', [...]

No, F(X) need not be a connected-abstract class even if we assume of X
not the disjunction but the conjunction, that X is *both* a connected
component and an abstract class.  Witness any full embedding F:G->H of
a group G (as a one-object groupoid) in a connected groupoid H having
more than one object.  Here F is an equivalence, and the set X of all
morphisms of G is both a connected component and an abstract class.
But not only is F(X) neither a connected component nor an abstract
class of H, it is not even a connected-abstract class of H.

What interesting theorem justifies adding "connected-abstract" to the
lexicon?

--
Vaughan Pratt
(FTPables: boole.stanford.edu:/pub/ABSTRACTS.)
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Name for this concept?
Date:         Tue, 02 Nov 93 12:51:22 SET
From:         Reinhard.Boerger@FernUni-Hagen.de

I think "ideals" should be closed under composition with all morphisms
(not just isos). I suggest the adjective "isomorphism-closed" or
"replete", which coincide with the common terminology for full
sucategories.
                      Reinhard Boerger
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Names of these concepts
Date: Tue, 02 Nov 1993 15:16:52 -0400 (EDT)
From: MTHISBEL@ubvms.cc.buffalo.edu

Max says he has called these globs 'ideals', and I wouldn't question that.
Except that he seems to say that Bill Lawvere has joined him in so calling 
them.
(Boy! Call me a taxi!)
Bill is not available at the moment. Steve Schanuel and I doubt that Bill has
called them ideals singularly or plurally. We suggest that Max and Bill have 
called a class of morphisms closed under pre- or postmultiplication by ANYBODY 
ideals. That is not Rowan's question. 
John Isbell
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Name for this concept?
Date: Tue, 2 Nov 1993 12:56:57 +0000
From: sjv@doc.ic.ac.uk (Steven Vickers)

>This rolls connectedness and abstractness into the one condition, as
>follows.  It had therefore better have some independently justifiable
>redeeming social value before burdening math dictionaries with its own
>name.

A category is a ring with many objects but no additive structure, and a
presheaf is then a left module. A representable presheaf is the ring
considered as a left module over itself (but there are lots of them because
of the many objects), and a sieve - a subpresheaf of a representable
presheaf - is a left ideal. The category is also a - just one - bimodule
(or profunctor) over itself just as a ring is, and the concept under
discusion, a subbimodule, is an ideal exactly as Max Kelly said (a 2-sided
ideal, or 2-sided sieve).

That's a justification by analogy with ring theory, though there is a gap:
ideals of rings are good not just because they are subbimodules. They are
also kernels of quotients, and once the additive structure is dropped then
it is no longer true that quotients are equivalent to subbimodules.

Steve Vickers.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Modules list
Date: Mon, 1 Nov 1993 23:03:45 -0800
From: "William H. Rowan" <rowan@crl.com>

Quite a number of people (about 30) responded to my proposed Beck modules
mailing list.  I now have an account that will support this.  However,
I have decided to do things a little differently.  The first change is to
broaden the scope a little bit.  It will now be about Beck modules,
their pointed set object counterparts, and in addition, ringoids.  Ringoids
are rings with several objects, or, small additive categories.  Ringoids
come into it naturally because the natural categories of A-modules, for
A a universal algebra, are identical with the categories of left modules
over certain ringoids, called the enveloping ringoids.  (I wrote my thesis
on this.)

The other change I am making is, instead of having a mailing list, to use
the Usenet news system, in particular the newsgroup sci.math.research.
This is a moderated newsgroup, the (slightly edited) announcement for which
follows:

  From dan@math.uiuc.edu Sat Oct  2 18:31:11 PDT 1993
  Article: 1298 of sci.math.research
  From: "Daniel R. Grayson" <dan@math.uiuc.edu>
  Newsgroups: sci.math.research
  Subject: Welcome to sci.math.research
  Date: Wed, 15 Sep 1993 07:02:22 -0500
  Organization: University of Illinois at Urbana
  Approved: Daniel Grayson <dan@math.uiuc.edu>
  Originator: dan@symcom.math.uiuc.edu
  
  Welcome to sci.math.research.  Here is our charter.
  
  CHARTER: This newsgroup is a forum for the discussion of
           current mathematical research.
  
  You are also encouraged to post announcements of
        - mathematics conferences
        - preprints available
        - new mathematics journals
        - online services for distribution of preprints
  
  A full archive maintained by Michael Boardman and Lake Forest College
  is available via gopher to math.lfc.edu, under the heading
        - Mathematics Related Items
  and its subheading
        - Archive of sci.math.research USENET newsgroup.
  It is indexed for fast retrieval of articles based on words within the text
  of the articles.
  
  Readers should also be aware of the mathematical information services
  available via gopher to e-math.ams.com.
  
  (end of announcement)

In order to help ourselves (and others) recognize messages to our subgroup,
we should all use the key phrase "Beck modules" (two words) in our subject
line, and also in the text of the message.  If you are new to the Usenet news
system, note that the threaded news reader trn will allow you to select only
those messages that have this key phrase, if that is what you want to do.

There are several clear advantages to doing this instead of a mailing list,
as well as a few disadvantages.  I find the Usenet news system to be quite
informative and convenient.  It is important to use a threaded news reader
such as trn, so that you will see discussion threads in sequence.  It is
_essential_ to learn how to use a news reader well so that you can filter
out stuff you don't want to read.  I imagine many of you have been doing this
for much longer than I have, and will be quite comfortable with it.  On the
other hand, if anyone has difficulty with this system (for example,
they have access to e-mail but not the newsgroup) please let me know and
I will try to help you stay in the loop somehow.

The discussion can begin at any time.  I screwed up this announcement the
first time, and no one got it.  In the mean time, I posted a so-called "FAQ"
to the news group to answer Frequently Asked Questions, and will post a new
version of it when I have accumulated some substantial changes, (and if and
when there is any discussion forthcoming!)

Note my new e-mail address: rowan@crl.com.  There is a place on this system
where I will be able to put files of common interest.

Bill Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Question about discrete opfibrations
Date: Wed, 3 Nov 93 17:28:07 +0100
From: Frank.Piessens@cs.kuleuven.ac.be (Frank Piessens)

Suppose D is a finite category. Does there exist a unique
discrete opfibration (dof) p:D->C such that C has a minimal number of objects?

(With "unique", I mean that any two dofs p:D->C and p':D->C', where C and
 C' have the minimal number of objects, are isomorphic in the sense that
 there exists an isomorphism i:C->C' such that p' = ip)

If so, is there an efficient algorithm to compute this discrete opfibration?

If not so, can you give a counter-example?

Thanks in advance for any replies,

Frank Piessens
Katholieke Universiteit Leuven.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Names of these concepts
Date: Wed, 3 Nov 93 16:52:56 +1100
From: kelly_m@maths.su.oz.au (Max Kelly)

Yes, of course, I misread the question in haste, thinking it referred to a
class of morphisms closed under left & right composition with any morphisms.
Sorry - Max Kelly.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Classes of arrows closed under isomorphism
Date: Thu, 4 Nov 1993 08:05:28 -0800
From: "William H. Rowan" <rowan@crl.com>

Many thanks for all the replies to my question about what to call a class
of arrows closed under composition with isomorphisms.  I now think a class of
arrows "closed under isomorphism" or "isomorphism closed" is a good 
terminology.
As for what good is it, it is true that if X is closed under isomorphism,
the image FX under a functor F is not necessarily closed under isomorphism,
but certainly FX generates such an isomorphism-closed class.  And, if F and F'
are naturally isomorphic, then FX and F'X generate the same isomorphism-closed
class.  This is very elementary, but I like it.

Bill Rowan
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: European Colloquium Category Theory -ECCT-
Note from moderator:

The following was posted on the Usenet Newsgroup sci.math.research yesterday. 
It is presumably of interest to subscribers to this list.
++++++++++++++++++++++++++++++++++++++++++++
From: damphous@univ-tours.fr
Date: Thu,  4 Nov 93 14:56:40 GMT

EUROPEAN  COLLOQUIUM  on  CATEGORY  THEORY  in  Tours  (France)

                        ----  ECCT ----

                       FIRST ANNOUNCEMENT
                      (November 4th 1993)

A  European Colloquium on Category Theory  (ECCT)  is  planned in TOURS
(France) from July 22nd to July 29th, the week before the International
Congress of Mathematics  in Z\"urich.  Mathematicians present in Europe
at that time are most welcome to pre-register now.  Having  an estimate
of the participation is important for us at this stage of the organiza-
tion
           To pre-register, you must proceed as follows:

           Register on the ECCT list:  send an e-mail  to
           ECCT-request@univ-tours.fr, with the following
           line in the body of the message  (do not put a
           subject in this e-mail).
           SUBSCRIBE

When pre-registering, you will automatically be sent  all pertaining
information  as the organization progresses. For any further inform-
ation, contact  Pierre Damphousse or Ren\'e Guitart at :
damphous@univ-tours.fr  or guitart@univ-tours.fr

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: e.e.c.t. and c.a.e.n. 94
Date: Wed, 10 Nov 93 18:08:44 +0100
From: ageron@univ-caen.fr (Pierre Ageron)

   Two events in France next summer might be of interest to category
theorists:

- the EUROPEAN COLLOQUIUM ON CATEGORY THEORY (E.E.C.T.)
(all areas of category theory)
in Tours, July 22-29
organizers: Pierre Damphousse and Rene Guitart
to pre-register: send the one-word message "SUBSCRIBE" to 
EECT-request@univ-tours.fr (no subject)


- the workshop CATEGORIES, ALGEBRES, ESQUISSES ET NATURALITES (C.A.E.N. 94)
(main themes: sketches, categories with structure, applications to logic
and algorithmics)
in Caen (Normandy), September 27-30
organizer: Pierre Ageron
to get more information: send a message to ageron@univ-caen.fr


    
   
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: category theory & linear optimization
Date: Wed, 10 Nov 93 17:03:46 PST
From: Michael J. Healy (206) 865-3123 <mjhealy@espresso.rt.cs.boeing.com>

I apologize in advance if this is not relevant or is already 
familiar, but there is work in using categorical methods for 
software synthesis to synthesize optimization algorithms and 
codes.  One reference for this is 

        D. R. Smith and M. R. Lowry (1990).  Algorithm Theories 
        and Design Tactics, Science of Computer Programming 14, 
        North-Holland, pp. 305-321.

                                        Regards,
                                        Mike Healy
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Why nothing beyond natural transformations?
Date: Thu, 11 Nov 93 11:18:42 GMT
From: jrk@sys.uea.ac.uk (Richard Kennaway)

I have a rather basic but vague question.

Categories, functors, and natural transformations seem to form the first
three elements of a series which could be continued indefinitely.  Why is
it that these three suffice, and that further members of the sequence are
almost never required?

Similarly, there are (1-)categories and 2-categories, and further members
of this sequence can be defined, yet they are rarely needed.  I have once
seen 4-categories referred to, but only once.

Is there some intuitive explanation for this?

--                                  ____
Richard Kennaway                  __\_ /    School of Information Systems
Internet:  jrk@sys.uea.ac.uk      \  X/     University of East Anglia
uucp:  ...mcsun!ukc!uea-sys!jrk    \/       Norwich NR4 7TJ, U.K.


++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Higher-dimensional categories
Date: Thu, 11 Nov 93 14:25:17 PST
From: baez@ucrmath.ucr.edu (john baez)

I'm very interested in the work of Louis Crane, Dan Freed and others
on how extending TQFTs to higher dimensions or higher codimensions
can be nicely phrased in the language of n-categories.  Luckily
James Dolan is here at UCR now and is educating me in such matters.
He and I are beginning to struggle towards a nice concept of
"weak n-categories," or even better "weak omega-categories," or perhaps
even better, "weak Z-categories" (in some sense a homotopical analog
of chain complexes).  I am interested in these things for doing physics,
but I dimly realize that lots of people have struggled with these concepts,
and I want to get a better grasp of what the key achievements and problems
in this subject are.  I have read Kapranov and Voevodsky's massive preprint
on Braided Monoidal 2-categories, 2-vector spaces and Zamolodchikov 
tetrahedra equations, and soon I should receive a copy of the new
paper on coherence in 3-categories by Gordon, Powell, and Street.  What
are the other main things I should find out about?  How come all you
categorists haven't yet invented a notion of "weak Z-categories," in
which there are n-morphisms for all integers n, and all relations
between n-morphisms are expressed in terms of (n+1)-morphisms?

Regards,
John Baez

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Why nothing beyond natural transformations?
Date: Fri, 12 Nov 1993 08:16:25 -0500
From: James Stasheff <jds@math.unc.edu>

the intuition behind those lacks 
may be lack of intuition (or good examples) on our part
but the times they are achanging
jim
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Question about discrete opfibrations
Date:         Fri, 12 Nov 93 13:22:14 SET
From:         Reinhard.Boerger@FernUni-Hagen.de

Let C be a category with 2 objects A,B and 2 morphisms u,v:A->B.
let D be the binary copower of C, i.e. the product of C with
the 2-object discrete category and let F:D->C be the codiagonal
funjctor mapping both copies of C identically to C. Let G:C->D
be the functor mpping one coy of C identically and interchanging
u and v in the other copy. Then F and G are opfibations, which
are not isomorphic under D (though they are isomorphic over C).

                                 Greetings
                              Reinhard Boerger
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Higher-dimensional categories (2 posts)
Date: Fri, 12 Nov 93 07:34:09 EST
From: barr@triples.Math.McGill.CA (Michael Barr)

John Baez and others interested in this question might begin by checking
out the work of Rainer Vogt.  Once when I was in Aarhus, he was telling
me that a weak infinity category was a simplicial set that satisfied
the Kan condition for interior faces only.  That is, for any collection
of n n-simplexes x^0,...,x^{i-1},x^{i+1},...x^n with 0 < i < n, such
that d^jx^k = d^{k-1}x^j, whenever j < k and none of j, k, k-1 was
i, there is an (n+1)-simplex x such that d^jx=x^j for j >< i.
Applied in dimension 2, this gives a weak composite.  In dimension 2
there is only one interior face and if this condition applied
uniquely, you would have a category with d^1x being the composite of
x^0 and x^2.  

Michael Barr
++++++++++++++++++++++++++++++++++++++++++
From: "Allen Knutson" <aknaton@and.Princeton.EDU>
Date: Fri, 12 Nov 93 9:14:40 EST

Have you looked at Quillen's "Homotopical Algebra"? I didn't much myself,
but it sounds potentially relevant, and Quillen's a mighty smart guy.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: job ad
From:   Richard Wood <rjwood@cs.dal.ca>
Date:   Wed, 3 Nov 1993 16:42:56 -0400

 Dalhousie University                   
 Department of Mathematics, Statistics and Computing Science
 Halifax, Nova Scotia
 Canada B3H 3J5
 phone (902) 494-2572
 FAX   (902) 494-5130
 
 
 
        Applications are invited at the Assistant Profesor level for two
 anticipated tenure track positions in the Division of Mathematics.  Duties
 include teaching at the Graduate and Undergraduate levels and maintenance 
 of a strong research programme.  Selection is based on demonstration of
 promise of excellence in research and teaching.  To apply, please send a
 curriculum vitae, selected reprints, and three letters of reference to
 
                Dr. R.P. Gupta, Chair
                Department of Mathematics, Statistics
                  and Computing Science
                Dalhousie University
                Halifax, Nova Scotia, Canada B3H 3J5
 
                phone: (902) 494-2572
                FAX:   (902) 494-5130
 
 The deadline date for applications is January 15, 1994.
 
 Dalhousie University is an Employment Equity/Affirmative Action Employer.
 The University encourages applications from qualified women, aboriginal
 peoples, visible minorities and persons with disabilities.  In accordance
 with Canadian  Immigration requirements, this advertisement is directed to
 Canadian Citizens and Permanent Residents.
 
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Why nothing beyond natural transformations?
Date: Sun, 14 Nov 93 15:18:58 +1100
From: street@macadam.mpce.mq.edu.au

>Date: Thu, 11 Nov 93 11:18:42 GMT
>From: jrk@sys.uea.ac.uk (Richard Kennaway)
>
>I have a rather basic but vague question.
>
>Categories, functors, and natural transformations seem to form the first
>three elements of a series which could be continued indefinitely.  Why is
>it that these three suffice, and that further members of the sequence are
>almost never required?

They don't, and they are. Lo, before the forties, everyone thought they
knew what "natural" was and thought not of defining it. Then came the
definition [E-ML] in terms of "categories". Categories themselves were 
seen as unnatural and unnecessary for many years; and still are by some.
As you have indicated, "2-categories" are now seen as quite natural
by many. To some of us, n-categories are as natural as n-simplexes and
n-cubes.

>Similarly, there are (1-)categories and 2-categories, and further members
>of this sequence can be defined, yet they are rarely needed.  I have once
>seen 4-categories referred to, but only once.
>
>Is there some intuitive explanation for this?

There may be a mathematical reason for feeling there is a barrier at
n = 3. It is easy to define n-categories and there are interesting
examples of these. But structural examples often form something less;
let's call them weak n-categories: the r-th composition is only
associative up to coherent (r-1)-equivalence etc. You may have met
weak 2-categories which are Benabou's "bicategories". Now every
bicategory is equivalent (in the approp sense) to a 2-category. 
Yet, not all weak 3-categories (Gordon-Power-St have called them 
tricategories) are equiv to 3-categories. Fear of the unknown is
natural; at the 3-level and above we are forced into more unfamiliar
and exotic phenomena. Exotic, yes, but not unnatural. 
This is where cubes and braids lurk, and who would call them unnatural?

Regards,
Ross



++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Higher-dimensional categories 
Date: Sun, 14 Nov 1993 07:24:24 -0500 (EST)
From: GLENN@CUA.EDU

I've been working for several months on an approach to higher
dimensional composition and the accompanying higher dim category
theory (functors, natural transformations, limits, etc...). (It may
be related to Vogt's work, cited in Barr's reply, but I'm unfamiliar
with that work.) I'd be happy to send a "work-in-progress" report (in
Latex format) giving details on what I've discovered thus far.

The "dimension 1" cases are ordinary categories. In the dimension n
case: objects are (n-1)-simplexes, maps are n-simplexes and
composites of maps are (n+1)-simplexes of a simplicial set C.

As Barr described in his response, the partial binary operation which
defines composition in an ordinary category, associates a map (=
1-simplex) to a pair of maps whose source and target match
appropriately.

In the n-dim case there is a partial (n+1)-ary operation which, given
n+1 n-simplexes whose faces match appropriately, produces their
"composite", another n-simplex whose faces match the given
ones according to the simplicial identities. One has to specify
*which* of the faces of the n+1 simplex is the composite of the
others. It can be the face opposite vertex i for any i = 0,..., n+1.

The most efficient way to specify all this is by saying that the
standard function from C_{n+1} ( = the n+1 simplexes of C) to the set
of open i-boxes of C of dimension n+1 is an isomorphism.

Examples (though special cases) of such structures are already known:
n-dimensional hypergroupoids. These include Eilenberg-MacLane spaces
and also arise from the singular complex of any topological space.
Rreferences to hypergroupoids include my 1982 paper ("Realization of
Cohomology Classes in Arbitrary Exact Categories", Jour. of Pure and
Appl. Alg. 25 (1982) 33-105) and papers by Jack Duskin.

There seems to be a reasonable "category theory" for this kind of
higher-dimensional composition. For example, functors C --> C' and
natural transformations of such functors occur as simplexes in the
function complex C'^C (of dimensions n-1 and n respectively). If C
and C' are higher dim categories in the sense I described above, then
so is C'^C.

Paul Glenn
Dept. of Mathematics
Catholic University of America
Washington, DC 20064
Internet: glenn@cua.edu  Phone: 202 319 5221
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Higher-dimensional categories 
Date: Mon, 15 Nov 93 10:11:01 +1100
From: street@macadam.mpce.mq.edu.au

I am very interested in seeing the paper of Paul Glenn, but in the
meantime, perhaps relevant is Conjecture 5.3 of my "The algebra of 
oriented simplexes" JPAA 49 (1987) 283-335. This is a slightly
more explicit form of a conjecture ( pre 1978 ) of John E. Roberts who 
motivated my work. Dominic Verity has proved this conjecture (it was 
announced in Bangor early this year and at the MSRI Conference last July).
Verity's preprint on this will be available very soon.
This work characterizes n-categories as simplicial sets with certain
elements at each dimension distinguished (and called "hollow" in
loc cit, but now we call them "thin" in accord with the Welsh School). 
The kinds of operations Barr and Glenn hint at are expressed as UNIQUE 
horn filler conditions. 

Regards,
Ross

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Three questions: on equalizers, e-mail to Manes, arrows as axioms
Date: Wed, 17 Nov 1993 06:23:37 +0100
From: France.Dacar@IJS.si

Greetings.

These days I'm going through a little categories-refresher
course.  Among my reading matter are the "Categories, Functors..."
by Arbib and Manes, and Manes' "Algebraic Theories".  My three 
questions are related to these.

Q1:  In "Algebraic Theories", there are several propositions
(say 4.22, 4.23, 4.24) which include the following assumption:
"Let (K,E,M) be a regular category _such that every equalizer is in 
in M_."  This mystifies me, because an equalizer is _always_ an 
M-monic, for any image-factorization system (E,M).  In any 
category, if an equalizer f factors as f = ge (composition to the 
left...) with e epi, then e is an isomorphism.  This is quite 
trivial; since I cannot imagine that this could have been 
overlooked for as long as image-factorization systems were around
up to the Manes' book, I am starting to doubt my ability to spot 
a mistake in my reasoning...

Q2:  How can I reach Manes or Arbib?  They're not on the 
structures mailing list.  I have collected quite an errata on 
their book which they still might find useful.

Q3:  This last question is about Universal Algebra vs. Monads and 
friends.  My knowledge of the relevant literature is rather
narrow, so this question has been probably adressed somewhere;
I'd like the relevant pointers.  As a kind of self-imposed exercise I
have tried to fill in the gap I felt there exists between the
old good techniquies of UA and that of high-flying categorial 
treatment involving monads and such.  I developed a little 
"elementary theory of UA in categories" which uses arrows as 
axioms, in the following sense.

Let C be any category, and A a set of arrows in C.  Say that an
object c of C _satisfies_ A if for every arrow p: a ---> a' in A,
any arrow f: a ---> c factors through p as f = f'p.  Denote by
C:A the full subcategory on all the objects that satisfy "axioms" A.
Then, playing around with conditions imposed on the category
C and on the axioms A, I can, step by step, reconstruct most of
the "standard" results of UA, up to and beyond the Birkhoff's
theorem.  All along the reasoning stays close to that in AU, just
translated into the language of arrows. 

At certain point a monad emerges, and then I can make 
the jump upwards to more rarefied regions.  For me this treatment 
by arrows as axioms provided the "missing link".
On the road up to the final ascent to monads there 
are plenty of points where one can go off in some other direction
and still get some benefit by transplanting part of UA techniques
"in abstract" to an un-algebraic matter.  One can also 
generalize:  there are other conditions on an object
expressible by a single arrow, one can allow restricted
boolean combinations of one-arrow conditions as basic statements
(say Horn implications), and so on.

If this was already done, where can I look it up?  If not, is it 
worth publishing, and where?

------------------------------------------------------------------------
France Dacar                            Email:    france.dacar@ijs.si
Computer Science Department             Phone:    +386 61 1-259-199 / 768
Jozef Stefan Institute                  Fax:      +386 61 1-258-058
Jamova 39, 61000 Ljubljana, Slovenia
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: higher dimensional groupoids
Date: Wed, 17 Nov 1993 17:48:37 +0000
From: mas010@bangor.ac.uk (Prof R. Brown)

Higher dimensional groupoids

One reason to see how higher dimensional categories exist and are useful is 
to look at the groupoid case. Higher dimensional groupoids form a useful 
higher dimensional version of groups and the fundamental group. The old 
result that "a set with two compatible group structures is  just an abelian 
group" (basically, 1932, with Cech's description of higher homotopy groups) 
was for long regarded as an obstruction to such a programme. In fact, sets 
with 2 compatible groupoid structures model 2-dimensional homotopy theory, 
and so on for n compatible groupoid structures. This makes them of course 
very complicated, even in dimension 2. This fact is interesting in itself. 

A clear problem was to define a higher homotopy groupoid. One solution was 
by the writer and PJ Higgins, as may be found in joint work in  (Proc London 
Math Soc 1978, for dimension 2, JPAA 1981 for all higher dimensions). This 
gives what we called the fundamental omega-groupoid of a filtered space. 
Later, Loday found a more complex and more general model, the fundamental 
cat^n-group of an n-cube of spaces. The existence of these functors is 
non-trivial. 

 My purpose in considering higher dimensional groupoids since about 1966 was 
the possibility of higher dimensional van Kampen Theorems, modelled on the 
groupoid version. The references are the above papers with Higgins, and also 
with J-L Loday (Topology, 1987, Proc London Math Soc 1987), with his more 
complicated algebraic model. These theorems and models now allow for 
specific computations of some homotopy invariants and even some homotopy 
types (my paper in the Adams Memorial Colloquium). The intuition is that one 
needs algebraic structures which can model the geometric notion of 
subdivision, and which have structure in a range of dimensions, to carry the 
information about how bits of a space fit together. Multiple groupoids (or 
categories) seem well placed for that.The intuition is related to old ideas 
in topology  of "What is a cycle?".  A cycle should be some kind of 
composition of the little pieces. How should one accomplish this, 
algebraically? One has to move away from a linear notation and an always 
defined composition. Taking the free abelian group on the little bits seems 
like a cop-out (but of course, it has its uses!).

One aspect of the theory is the equivalence of various views of a given 
structure. This is referred to by Ross Street in his communication, with 
regard to infinity-categories. So we have a set of equivalences between 
crossed complexes, omega-groupoids (RB-PJH, JPAA 1981), infinity-categories 
(CTGDC 1981), cubical T-complexes (CTGDC 1981), simplicial T-complexes 
(Ashley's thesis of 1978, published in Diss Math 155, 1988), polyhedral 
T-complexes (Jones, 1984, published as Diss Math 156, 1988). Here T-stands 
for "thin". These canonical structures are a generalisation of identities 
which appear only from dimension 2. The idea appears in old axioms for 
groups; with Keith Dakin in Bangor in 1975; and independently with Roberts 
in Australia at the same time, with the name "hollow". Thin elements play a 
crucial role in computations in and manipulations with these objects. 
Intuitively, thin elements have commutative boundary, and commutative 
boundaries have thin fillers. Thin elements also define compositions: all 
faces but one of a thin element "compose" to give the remaining face. For 
more on the philosophy of this, see Jones' thesis, which deals with 
compositions of general faces. 

Even more complex is the equivalence between cat^n-groups and crossed 
n-cubes of groups (Ellis-Steiner, JPAA, 1987). 

These equivalences allow a translation from a linear notation to a "higher 
dimensional" notation. Verity's equivalence in the category case mentioned 
by Ross Street, is in the same spirit as, and generalises, the equivalence 
between simplicial T-complexes and infinity-groupoids (go through crossed 
complexes, combining RB-PJH with Ashley).  

These things work. I just realised that one can do many sums with results of 
the first paper of RB-PJH, which for example allows the computation not only 
of \pi_2(BH \cup CBG) when G is a subgroup of H, for specific G,H, but also 
of the first k-invariant, i.e. the 2-type, of this space. 

These ideas are all modelled on groupoids, as is natural for the homotopy 
applications and the  Generalised Van Kampen Theorems. A categorical version 
has not been done in full generality (polyhedrally). One does not expect the 
same type of application. There are relations with cohomology and knot 
theory which need more investigation. A lot of these ideas were discused in 
the visit of Ross and Dominic Verity to Bangor in May/June, 1993.

 Ronnie Brown

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Three questions: on equalizers, e-mail to Manes, arrows as axioms
Date: Wed, 17 Nov 1993 18:46:40 -0800
From: "William H. Rowan" <rowan@crl.com>

Concerning axiomatization of universal algebra using arrows, usually
we consider that an identity in universal algebra is given categorically
by giving a pair <f,g> of elements of a free algebra F (or, a pair of
homomorphisms from a free algebra on one element into F) and then an
algebra X satisfies <f,g> iff all homomorphisms F->X coequalize <f,g>.
I think this is explained sketchily in MacLane, CWM.  This is subsumed by
your framework because we only need to form the coequalizer h:F->G of <f,g>
and then take {h} as your class A of arrows.  Perhaps your framework has
greater expressive power than the usual one in universal algebra, in which
case it might be worth publishing, but I can offer no strong opinion to that
effect, and indeed you will probably have to figure this out yourself.  You
also need to consider that perhaps someone else has an equivalent or even
more general framework which is all worked out, such as sketches or something.
Perhaps some other people will have other comments.

Bill Rowan
+++++++++++++++++++
Date: Thu, 18 Nov 93 10:07:16 CST
From: strecker@math.ksu.edu (George Strecker)


 Let C be any category, and A a set of arrows in C.  Say that an
 object c of C _satisfies_ A if for every arrow p: a ---> a' in A,
 any arrow f: a ---> c factors through p as f = f'p.  Denote by
 C:A the full subcategory on all the objects that satisfy "axioms" A.

I believe that this was first done by Banaschewski and Herrlich
Houston Math. J. (1977) 149-171.

See also Section 22 of the book "Abstract & Concrete Categories"
[Wiley Interscience].

George Strecker

++++++++++++++++++++++
From: koslowj@math.ksu.edu (Juergen Koslowski)
Date: Thu, 18 Nov 93 9:20:56 CST

Concerning question 1 of France Dacar: 

> Q1:  In "Algebraic Theories", there are several propositions
> (say 4.22, 4.23, 4.24) which include the following assumption:
> "Let (K,E,M) be a regular category _such that every equalizer is in 
> in M_."  This mystifies me, because an equalizer is _always_ an 
> M-monic, for any image-factorization system (E,M).  In any 
> category, if an equalizer f factors as f = ge (composition to the 
> left...) with e epi, then e is an isomorphism.  This is quite 
> trivial; since I cannot imagine that this could have been 
> overlooked for as long as image-factorization systems were around
> up to the Manes' book, I am starting to doubt my ability to spot 
> a mistake in my reasoning...
> 

I don't have my copy of Manes' AT handy, but it seems to me that the
assumption that E consists of epis is not warranted. After all, 
E = all morphisms and M = all isos constitute an image-factorization
system, and not all equalizers are isos.

Regards, J"urgen 
-- 
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)

+++++++++++++++++++++
Date: Thu, 18 Nov 93 11:20:58 -0500
From: cfw2@po.CWRU.Edu (Charles F. Wells)


The work described in Point 3 of France Dacar's recent message
looks very much like a special case of work of Andreka and
Nemeti and Guitart and Lair.  The basic references are given
below in BibTeX input form, which is (I fondly believe)
self-explanatory:

@ARTICLE{
author = "Andreka, H. and I. Nemeti",
title = "Formulas and ultraproducts in categories",
journal = "Beit. zur Alg. und Geom.",
volume = 8,
year = 1979
}

@ARTICLE{
author = "Guitart, Ren\'e and Christian Lair",
title = "Calcul Syntaxique des Mod\`eles et Calcul des Formules Internes",
journal = diagrammes,
year = "1980",
volume = "4",
}

I do not have the Andreka and Nemeti article.  These articles
are also relevant:

@ARTICLE{
author = "Guitart, Ren\'e and Christian Lair",
title = "Limites et Co-limites pour
Repre\-senter les For\-mu\-les",
journal = diagrammes,
year = "1982",
volume = "7",
}

@ARTICLE{
author = "Guitart, Ren\'e",
title = "On the Geometry of Computations (I)",
journal = cahiers,
year = "1986",
volume = "27",
pages = "107--136",
}


--
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: Arrows as Axioms
Date: Fri, 19 Nov 93 18:56:34 +0100
From: momathie@mathp7.jussieu.fr (Monique Mathieu)

 Ce dont parle F.Dacar est largement etudie dans :

1.  H. Andreka et I. Nemeti ,
"Generalization of the Concept of Variety and Quasi-Variety to
Partial Algebras through Category Theory"
Diss. Math. CCIV , Warszawa , 1983.

On peut en trouver une "version modernisee" dans :

2.  F. Cury ,"Completion et Completude selon H.Andreka. et I.Nemeti.
Diagrammes 29 , Paris 1992.

Le cas particulier de "il existe un unique" se touve dans :

3. L. Coppey, Theories Algebriques et Extensions de Pre-faisceaux"
Cahier de Top. et de Geometrie Diff. , Vol. XIII - 1 et XIII - 4 ,
Paris 1972 ,

4.  L. Coppey et C. Lair , "AMEN, Algebricite - Monadicite -
Esquissabilite et Non-algebricite ,
Diagrammes 13 , Paris 1985.

Anterieurement, H. Andreka, I. Nemeti et I. Sain avaient meme
etendu la satisfaction (par un objet) d'un ensemble A de fleches en la
satisfaction d'un ensemble de familles projectives de fleches. Voir,
par exemple :

5.  A. Andreka et I. Nemeti , "A General Axiomatizability Theorem
formulated in terms of Cone Injective Subcategories" ,
Coll. Math. Soc. , J. Bolyai  29 , Univ. Alg. Esztergon (Hungary)
1977.
6.  I. Nemeti et I. Sain , "Cone Implicational Subcategories and
some Birkhoff-Type Theorems"
Coll. Math. Soc. J. Bolyai 29 , Univ. Alg. Esztergon (Hungary)
1977.
7.  H. Andreka et I. Nemeti , "Los Lemma holds in every Category"
Studia Scientiarum Mathematicarum Hungarica 13 , 1978.
8.  H. Andreka et I. Nemeti , "Injectivity in Categories to represent
all First Order Formulas" ,
Demonstratio Math. Vol XII - 3 , 1979.

L'aspect completement categorique des choses consiste a satisfaire
des ensembles de cones projectifs (d'indexations non necessairement
discretes). Ceci a ete traite originellement (a la suite des travaux de
H. Andreka, I. Nemeti et I. Sain) tout d'abord par :

9.  R. Guitart et C. Lair , "Calcul Syntaxique des Modeles et Calcul
des Formules Internes" ,
Diagrammes 4 , Paris 1980.

Le lien systematique avec les categories de modeles d'esquisses
(i.e. de theories du premier ordre) est largement explicite par :

10.  C. Lair , "Categories Qualifiables et Categories Esquissables"
Diagrammes 17 , Paris 1987.

Ces derniers temps, certains reprennent les idees qui se trouvaient
dans 1.  pour, vraisemblablement, aboutir dans quelques annees
a celles qui se trouvent dans 10??

11.  J. Adamek et J. Rosicky , "Locally Presentable and Accessible
Categories"
Livre a paraitre.
12.  H. Makkai , "Generalized Sketches as a Framework for
Completeness Theorem"
Pre-print , 1993.

Monique Mathieu, Universite Paris 7, Denis Diderot.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: normed categories
Date: Sun, 21 Nov 93 07:26:51 CST
From: koslowj@math.ksu.edu (Juergen Koslowski)

In "METRIC SPACES, GENERALIZED LOGIC, AND CLOSED CATEGORIES" (Rend.
del Sem. Mat. e Fis.  di Milano 43 (1973)) Bill Lawvere suggests the
notion of a "normed category" as a category enriched in a suitable
(symmetric monoidal) closed category S(R). (Here R denotes the
interval [0,\infty], ordered by >= and with + as tensor and truncated
- as internal Hom.)

Has anybody worked out the details? 

Presumably, the objects of S(R) are to be subsets of R. This is
suggested by the claim that there ought to be a closed functor inf
from S(R) to R that induces the passage from normed categories to
metric spaces (and by the use of the same symbol "S" that is used to
denote the category of sets). As with any commutative monoid, the
power-set of R under inclusion carries a symmetric monoidal closed
structure: A + B = { a+b | a\in A and b\in B } (point-wise) defines
the tensor and C - A = { x\in R | A + {x} contained in C } (not
point-wise!) defines the internal Hom. {0} is the unit object.
Clearly, inf turns into a strong functor as required.

But how am I to interpret "the fundamental property of a normed
category" (top of p. 140), namely (*) |f| + |g| >= |fg|? This
inequality seems to apply to ordinary (= Set-enriched) categories X
that carry an extra structure, namely a function |-| from Mor(X) to
R. Presumably, this function also ought to satisfy (**) 0 >= |id_Y|
for any X-object Y. E.g., one could define |f| to be 0 for every
isomorphism of X, and 1 for every other morphism. In general this does
NOT yield an S(R)-enriched category in the sense defined above. In
fact, the definition above only seems to support the equality |f| +
|g| = |fg|.

Can further morphisms be added to S(R) without destroying the monoidal
closed structure? Yes, if A +{q} is contained in B one can interpret q
as a morphism from A to B. (The original inclusions arise for q = 0.)
All these new morphisms f satisfy x <= f(x). But in view of the
preceding paragraph one would prefer to have morphisms g that satisfy
x >= g(x).  Since this latter condition does not allow any new
morphisms with domain {0}, the monoidal closed structure cannot be
maintained. I must be missing something obvious.  E.g., the definition
of S(R) above doesn't use the symmetric monoidal closedness of R
itself at all. Could someone please point me in the right direction?

Thank you!

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)
aberne@dhvrrzn1.uni-hannover.d400.de
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: normed categories
Date: Sun, 21 Nov 1993 10:42:22 -0500 (EST)
From: MTHFWL@ubvms.cc.buffalo.edu

No,it is not the power set of a closed category in which the normed categories
are enriched, but rather the portion of the presheaf category (closed via 
Brian
Day) consisting of coproducts of representables, for which there is an
alternative description of the closed structure. Some details of this
construction were published in a paper by Betti and Galluzzi. It is also
mentioned in the introduction to Springer Lecture Notes 274 that real numbers
are merely the poset reflection of the category of dynamical systems.  -Bill
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Arrows as Axioms
From: Jiri Rosicky <rosicky@math.muni.cz>
Date: Mon, 22 Nov 1993 14:15:41 +0100 (MET)

One more reference added to the list of Monique Mathieu:
11a. J.Adamek, J.Rosicky: On injectivity in locally presentable categories,
     Trans. Amer. Math. Soc. 336 (1993), 785-804
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Sydney ftp site: current contents
Date: Wed, 24 Nov 1993 18:12:58 +1100
From: walters_b@maths.su.oz.au (Bob Walters)

                               SYDNEY  
             CATEGORIES COMBINATORICS and COMPUTER SCIENCE
                                CCC

                 Material Available by Anonymous FTP
                          from maths.su.oz.au
                            129.78.68.2

========================================================================    
This file is README in the sydcat directory of maths.su.oz.au,
129.78.68.2, accessible by anonymous ftp.

The sydcat directory is for FTP distribution of recent publications, 
programs, seminar listings and other material of the 

Sydney Category Theory, Combinatorics, and Computer Science Group.
========================================================================    
This group consists of staff and students at the 
        University of Sydney
        Macquarie University
        University of New South Wales
        University of Technology Sydney
Sydney, Australia, including the following:

        Murray Adelman                  murray@macadam.mpce.mq.edu.au
        Brian Day
        Lee Flax                        flax@macadam.mpce.mq.edu.au
        Robbie Gates                    gates_r@maths.su.oz.au
        Amitavo Islam                   islam_a@maths.su.oz.au
        Barry Jay                       cbj@socs.uts.edu.au
        Mike Johnson                    mike@macadam.mpce.mq.edu.au
        Giulio Katis                    katis_p@maths.su.oz.au
        Max Kelly                       kelly_m@maths.su.oz.au
        Mark Leeming                    leeming_m@maths.su.oz.au
        Stephen Ma                      ma_s@maths.su.oz.au
        Wafaa Khalil                    khalil_w@maths.su.oz.au
        Wesley Phoa                     wes@cs.unsw.oz.au
        Usha Sridhar                    sridhar_u@maths.su.oz.au
        Ross Street                     street@macadam.mpce.mq.edu.au 
        Sun Shu-Hao                     sun_s@maths.su.oz.au
        Dominic Verity                  domv@macadam.mpce.mq.edu.au 
        Henry Weld                      weld_h@maths.su.oz.au
        Bob Walters                     walters_b@maths.su.oz.au
        Karl Wehrhahn                   wehrhahn_k@maths.su.oz.au

=======================Available material===============================    
The following files and directories are available:

ADDRESS lists are in sydcat/addresses
BIBLIOGRAPHY files are in sydcat/bibliography
BOOK descriptions are in sydcat/books
PAPERS:dvi files of papers of author are in sydcat/papers/author
CONFERENCE details are in sydcat/conference
SEMINARS - some details are in sydcat/seminars
SOFTWARE - programs are in sydcat/software

Here is the current list of contents:

addresses/catcurrent    
        The  Category Theory address list 
        maintained by Max Kelly and Michael Johnson
        Updated regularly   

addresses/structdir     
        Vaughan Pratt's email address list
        Updated October 1993   
        Release 3.0
        Master copy: Boole.Stanford.EDU:~ftp/pub/struct.dir
        Maintainer: Vaughan Pratt, pratt@cs.stanford.edu

bibliography            
        A directory for bibliographies: contains some
        Bibtex files of sydcat publications

bibliography/reports    
        List of Research Reports of the Pure Mathematics
        Department, University of Sydney, Australia
        Updated regularly.

books/hibi
        Contents and bibliographic details of
        Takayuki Hibi 
        "Algebraic combinatorics on convex polytopes"
        Carslaw Publications
        PO Box 615
        Glebe, NSW 2037
        Australia

books/walters
        Information about, corrections to,
        and bibliographic details of 
        R.F.C. Walters 
        "Categories and Computer Science"
        published by Carslaw Press in Australia
        and by Cambridge University Press elsewhere

        Contents and bibliographic details of
        RFC Walters 
        "Number theory an introduction"
        Carslaw Publications
        PO Box 615
        Glebe, NSW 2037
        Australia

books/wehrhahn
        Contents and bibliographic details of
        KH Wehrhahn 
        "Combinatorics: an introduction"
        Carslaw Publications
        PO Box 615
        Glebe, NSW 2037
        Australia

conferences            
        Details of conferences

papers/jay
        A directory containing papers of Barry Jay

        Title: Compositional Characterization of Observable Program Properties
        Authors: B. Steffen,C. Barry Jay, M. Mendler
        Process: Aarhus TR and accepted by Journal

        Title:  Coherence in Category Theory and the Church-Rosser Property
        Author:CBJ
        Process: published in Studia Logica

        Title: Modelling reduction in confluent categories
        Author: CBJ
        Process: Published in the Durham Proceedings

        Title: Fixpoint and Loop Constructions as Colimits
        Author: CBJ
        Process: Como Proceedings

        Title:Long $\beta\eta$ normal forms and confluence (revised)
        Author: CBJ
        Process: static cf virtues.tex

        Title: Extending properties to categories of partial maps
        Author: CBJ
        Process : static

        Title: Partial Functions, Ordered Categories, Limits and Cartesian 
Closure
        Author: CBJ
        Process: proceedings of the HOW 

        Title: Tail recursion through universal invariants
        Author: CBJ
        Process: to appear in TCS

papers/johnson
        M.S.Johnson
        Linear term rewriting systems are higher dimensional 
        string rewriting systems,
        {\em Proc of the Institute for Mathematics and its Applications\/} 
        (1991), 101--110.
        
        M.S.Johnson
        (with G.P.~Monro and C.N.G.~Dampney) 
        A mathematical foundation for ERA, 
        {\em Proc of the Institute for Mathematics and its Applications\/} 
        (1991), 77--84.
        
        M.S.Johnson
        (with R.F.C.~Walters) 
        Category theoretic modelling of digital circuits and systems,
        {\em Proc of the Pan-Commonwealth Conference on Mathematical 
Modelling}
        {\em in Circuit Design}, Commonwealth Science Council (1992), 
199--213.
        
        M.S.Johnson
        (with R.F.C.~Walters) 
        Algebra objects and algebra families for finite limit theories,
        {\em Journal of Pure and Applied Algebra} 
        {\bf 83} (1992) 283--293.

        M.S.Johnson
        (with S.-H. Sun) 
        Remarks on representations of universal algebras by sheaves of 
        quotient algebras, M.S.Johnson
        {\em Proceedings of the Canadian Mathematical Society\/} 
        {\bf 13} (1992) 299--307.

        M.S.Johnson
        (with C.N.G.~Dampney and P.~Deuble)
        Taming large complex information systems, M.S.Johnson
        {\em Proceedings of Complex Systems} '92, IOS Press,
        Amsterdam, 210--222.
        
        M.S.Johnson
        (with R.~Buckland)
        An application of logic programming in pure mathematics,
        to appear in {\em Proceedings of ACSC}, 1993.
        
        M.S.Johnson
        (with C.N.G.~Dampney)
        Category theory and information systems engineering,
        {\em Proceedings of AMAST93}, Unviersity of Twente, Holland, 95--103.
        
        M.S.Johnson
        On the value of commutative diagrams in information modelling,
        to appear in {\em Springer Workshops in Computing}, 16 pages.

papers/kelly    
        G.M. Kelly, Stephen Lack: Finite-product-preserving functor, 
        Kan extensions, and strongly-finitary 2-monads

        G.M. Kelly, Stephen Lack, R.F.C. Walters,
        Coinverters in categories with structure

        galaus.dvi    
        G.Janelidze and G.M.Kelly, 
        Galois theory and a general notion
        of central extension. (A4 version)   

        galam.dvi
        G.Janelidze and G.M.Kelly, 
        Galois theory and a general notion
        of central extension. (American quarto version)   

        janmark.dvi   
        G.Janelidze and L.Ma'rki, 
        Radicals of ringa and pullbacks.

papers/phoa
        Wesley Phoa
        bohm.ps
        "From term models to domains"
        --describes a category of `synthetic domains'
        for the closed term model in which
        terms with the same Bohm tree are identified

        Wesley Phoa
        graph.ps    
        "Building domains from graph models"
        --describes a category of `synthetic domains'
        in the realizability topos arising from
        the r.e. graph model of the lambda-calculus 

        Wesley Phoa
        pcf.ps      
        "A note on PCF and the untyped lambda-calculus"
        --proof of computational adequacy of an untyped
        translation of call-by-name PCF

        Wesley Phoa
        poly.ps     
        "A simple categorical semantics for first-order
        polymorphism"
        --describes how any cartesian closed category can
        be used to model ML polymorphism, using the
        notion of `polynomial category'

        M.P. Fourman and Wesley Phoa
        sml.ps      
        "A proposed categorical semantics for Pure ML"
        (with M. P. Fourman, LFCS)
        --sketch of a semantics for SML using synthetic
        domain theory; focuses on the Modules system

        Wesley Phoa
        subtypes.ps 
        "Using fibrations to understand subtypes"
        --informal account of categorical models for  
        subtyping and bounded quantification

        Wesley Phoa
        synth.ps    
        "Effective domains and intrinsic structure"
        --describes a category of `synthetic domains'
        in the effective topos

        Wesley Phoa
        tech.ps     
        Replacing fibs.ps, topoi.ps, eff.ps,
        these notes provide an introduction to (some aspects of)
        a)  fibrations and polymorphic lambda calculus
        b)  constructive logic, categorical logic and topos theory
        c)  Kleene realizability; PERs and omega-sets
        d)  the effective topos; modest sets and how they model polymorphism
        They assume some basic knowledge of category theory, logic and typed
        lambda calculus.  No familiarity with indexed categories or with 
        categorical logic or topos theory is required.
        The notes do not attempt to be comprehensive, but simply try to give
        a reasonably relaxed account of the material.  They are about 150pp
        including the index and appendixes.  There are plenty of exercises.

papers/sun
        Shu-Hao Sun, RFC Walters, 
                Representations of modules and cauchy completeness

        Shu-Hao Sun, 
                Adjunction of the associated sheaf functor of non-commutative 
rings

        Shu-Hao Sun,
                Non-commutative Deligne formula

        Shu-Hao Sun,
                Biregular rings and their duality

        Shu-Hao Sun,
                Non-commutative quasi-coherent sheaves
        
        Shu-Hao Sun,
                Equivalence of algebraic and geometric local cohmology

        Shu-Hao Sun,
                Structure sheaves for non-commutative rings

        Shu-Hao Sun,
                Duality on compact prime ringed spaces
        
        Shu-Hao,
                Generalized Grothendieck topologies

papers/walters
        Aurelio Carboni, Stephen Lack, R. F. C. Walters, 
                An introduction to extensive and distributive categories, 
Jun92.

        S. Carmody, R.F.C. Walters, 
                The Todd-Coxeter Procedure and Left Kan Extension, Mar91
        
        S. Carmody, R.F.C. Walters, 
                Computing quotients of actions of a free category, Mar91.

        M. S. Johnson, R.F.C. Walters, 
                Algebra Families, Apr92

        P. Katis, N. Sabadini, RFC Walters, 
                On discrete dynamical systems and concurrency, Dec93

        G.M. Kelly, Stephen Lack, R.F.C. Walters,
                Coinverters in categories with structure, Jun92

        Wafaa Khalil, R.F.C. Walters, 
                An imperative language based on distributive categories II 
Apr92
        
        Wafaa Khalil, R.F.C. Walters, 
                Functional processors and operations on them 
                in extensive categories, May93
        
        Wafaa Khalil, Eric Wagner, R.F.C. Walters,
                Fix-point semantics for programs in distributive
                categories, May93
        
        Mark Leeming, R.F.C. Walters,
                Computing left kan extensions using the Todd-Coxeter
                procedure

        N. Sabadini, S. Vigna, RFC Walters
                An automata-theoretic approach to concurrency
                through distributive categories: on morphisms, Jan93

        N. Sabadini, R.F.C. Walters
                On functions and processors:
                an automata-theoretic approach to concurrency
                through distributive categories, Nov92

        N. Sabadini, H. Weld, R.F.C. Walters, 
                Distributive automata and asynchronous circuits, May93

        N. Sabadini, S. Vigna, RFC Walters
                A notion of refinement for automata, Jan93
        
        Shu-Hao Sun, RFC Walters, 
                Representations of modules and cauchy completeness, Mar93

        R.F.C. Walters, 
                An imperative language based on distributive categories, 91

seminars/sydcat         
        sydcat.tex is a listing of seminars given at the
        Sydney Category Seminar
        Not being currently maintained.

seminars/cics
        cics.tex is a listing of seminars given at the
        Sydney Categories in Computer Science Seminar.
     
software/kan_1.0        
        kan (vers 1.0)  (Sean Carmody, Craig Reilly, Bob Walters)
        An implementation of the algorithm developed in 1990 by 
        Carmody & Walters to compute (finite) left Kan extensions 
        is now operational (programmed by Reilly and Carmody).

        There are also some sample input files as well as a file 
        called KAN.info which gives further details of the program
        and one called README which describes the sample input files. 
        If you experiment with the program, we would be very interested 
        to hear any comments or suggestions (especially with regards 
        any bugs which you -- hopefully won't -- find).
                                                 
        Future versions of the program will include a more 
        standardised i/o format, and will renumber the elements of 
        the output sets will be of the form {1,2,..,n} (a set which 
        may currently be given as {1,3,7} would become {1,2,3}).

        Sean Carmody.
        email: carmody_s@maths.su.oz.au
        21 May 1991

software/kan_2.0        
        A later version of kan, but Sean left Sydney for Cambridge
        before it was properly documented.

software/buckland       
        Representing pasting schemes in Prolog
        A program written by Richard Buckland, with Michael Johnson.
        For details see programs/buckland/readme
        Further details email 
        t-richbu@microsoft.com
        mike@macadam.mpce.mq.edu.au

--
Bob Walters
Department of Pure Mathematics, University of Sydney, NSW 2006, Australia
Internet: walters_b@maths.su.oz.au  Phone: +61 2 692 2966  FAX: +61 2 692 4534

==========================Instructions======================================

FTP LOGIN.  Give the following commands.

        ftp maths.su.oz.au
Login:  anonymous                       (if you don't have an account on 
maths)
Paswd:  yoursurname                     (though any string will work)
        bin                             (if you are retrieving a .dvi file)
        prompt off                      (if you want no ? prompts from mget)
        cd sydcat                       (change directory to _public/sydcat
        ls -lt                          (see what's there, most recent first)
        mget filename-1 ... filename-n  (e.g.   mget catcurrent.Z)
        quit                            (exit from FTP)


DVI.  If you wish to print paper, calg say, retrieve calg.dvi and
associated .eps and .sty files from the subdirectory calg (cd calg first).
You must first give the bin command to ftp since .dvi files are not 
text files.  You will then need a dvi to postscript converter
which will include the .eps files. Print the resulting postscript file
on your host.

PROBLEMS.  If you have problems in either retrieving or compiling
papers, please contact Bob Walters.

NOTE.  Please note that the IP satellite link between Australia and the rest 
of the
world is saturated most of the time.  Large file transfers to non-Australian
sites should be spaced out, and should preferably take place between the
hours 2300 and 0800 local Eastern Australian time (the local time appears in
the ftpd banner at connection).

============================================================================   
 

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Initial model of higher sketches
Date: Fri, 26 Nov 93 19:00:04 +0900
From: Hiroyuki Miyoshi <miyoshi@slab.sfc.keio.ac.jp>


Does anyone know results about the existance of initial model (in Cat
or other categories) of 2-sketch or V-sketch with indexed limits, and
that of other equivalent (or more general) formulations (e.g. 
higher-dimensional sketch of Power and Wells)?

I think of results analogous to the existance of the (family of)
initial model in Set of FL or FLS sketch.  Any relevant informations
and references are much welcome.

Thanks in advance.

----
Hiroyuki Miyoshi
Department of Mathematics, Faculty of Science and Technology, Keio Univ.
OFFICE: c/o Nobuo Saito, Faculty of Environmental Information, Keio Univ.
        5322 Endoh, Fujisawa 252, JAPAN
EMAIL:  miyoshi@slab.sfc.keio.ac.jp
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Initial model of higher sketches
Date: Fri, 26 Nov 93 20:39:37 EST
From: otto@triples.Math.McGill.CA (James Otto)

>Does anyone know results about the existance of initial model (in Cat
>or other categories) of 2-sketch or V-sketch with indexed limits, and
>that of other equivalent (or more general) formulations (e.g. 
>higher-dimensional sketch of Power and Wells)?
>
>I think of results analogous to the existance of the (family of)
>initial model in Set of FL or FLS sketch.  Any relevant informations
>and references are much welcome.
>
>Thanks in advance.
...
>Hiroyuki Miyoshi
...
>EMAIL: miyoshi@slab.sfc.keio.ac.jp

i'll overly busy until the end of '93, but here's some thoughts.  you
can probably describe (e.g. see my thesis, summer '94) the modeling of
your V-sketchs in appropriate V-cats as set models of an FL sketch,
or, more humanely, of an essentially algebraic presentation.  then the
initial Herbrand model of that gives you an initial V-cat model.
actually there is some fiddling about approximating pseudo maps by
strict maps, e.g.  approximating non locally finitely presentable cats
by locally finitely presentable ones.  e.g. see the Australians on
flexible limits.  also M. Makkai, here at McGill, has a view of
sketches in which weak initial models are seen as injective hulls.

bon soir, J. Otto
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Initial model of higher sketches
Date: Mon, 29 Nov 93 14:34:35 +1100 
From: kelly_m@maths.su.oz.au (Max Kelly)

Miyoshi might be interested in my article "Structures defined by finite limits
in the enriched context I ", Cahiers de Topologie et Ge'om. Diff. 23(1982),
3-42.

Max Kelly.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
Date: Mon, 29 Nov 93 09:52:49 EST
From: stiller@blaze.cs.jhu.edu

Is there a categorical analogue of the following well-known matrix
identity: If A, B, C, D are matrices over a field and juxtaposition
denotes matrix multiplication and \otimes denotes Kronecker product,
then

(A\otimes B)(C\otimes D)=(AC)\otimes (BD)

whenever the dimensions are consistent?

Note: I am a category theory novice, but the proof of this identity
uses so few of the field axioms, and is so useful, (and curiously
looks like the interchange of horizontal and vertical compositions of
natural transformations which I presume is coincidental) that I was
just curious if an analogue of this identity is true in more general
categories than matrices over rings.
-- 
Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University.  
stiller@cs.jhu.edu. "Tertan I am, but what is Tertan?  Of this time, of 
that place, of some parentage, what does it matter?"


++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
Date: Mon, 29 Nov 93 15:56:21 PST
From: baez@ucrmath.ucr.edu (john baez)

Lewis Stiller writes:
 
Is there a categorical analogue of the following well-known matrix
identity: If A, B, C, D are matrices over a field and juxtaposition
denotes matrix multiplication and \otimes denotes Kronecker product,
then
 
(A\otimes B)(C\otimes D)=(AC)\otimes (BD)
 
whenever the dimensions are consistent?
 
Note: I am a category theory novice, but the proof of this identity
uses so few of the field axioms, and is so useful, (and curiously
looks like the interchange of horizontal and vertical compositions of
natural transformations which I presume is coincidental) that I was
just curious if an analogue of this identity is true in more general
categories than matrices over rings.
 
----------
I am just learning what some of the people on this group invented, so
only the enthusiasm of a novice can justify my attempt to explain this.
Briefly, this "exchange identity" (or maybe it's called "interchange")
is indeed symptomatic of a very general phenomenon in category theory,
and the analogy with horizontal and vertical compositions is NOT
coincidental.  A 2-category is, roughly, a category in which homsets
are categories and composition is a (bi)functor.  One always has
an exchange identity in such a structure.  The category of categories
is a 2-category, indeed the primordial one.  A monoidal category (a
category with tensor products satisfying nice axioms) can be viewed
as a 2-category in such a way that your exchange identity becomes a
special case.  Another nice example is the 2-category in which
objects are maps from a point into a topological space, morphisms
are maps from a unit interval, and morphisms-between-morphisms (so-called
2-morphisms) are maps from a square.  Well, here associativity does
not hold "on the nose" but only "up to reparametrization" so we
are touching upon the great puzzle of "weak n-categories".  In any 
event, this situation provides a very nice geometrical interpretation
of the exchange identity.  

jb

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
Date: Tue, 30 Nov 93 15:11:36 +1100
From: street@macadam.mpce.mq.edu.au

>Is there a categorical analogue of the following well-known matrix
>identity: If A, B, C, D are matrices over a field and juxtaposition
>denotes matrix multiplication and \otimes denotes Kronecker product,
>then
>
>(A\otimes B)(C\otimes D)=(AC)\otimes (BD)
>

There is a (skeletal) category Mat whose objects are finite ordinals,
whose arrows n --> m are mxn matrices, and whose composition is matrix
multiplication. Mat becomes a monoidal category with Kronecker product
as its tensor product. Your equation expresses the functoriality of this
tensor product  Mat x Mat ---> Mat  which is also strictly associative.

The category Vect of finite dimensional vector spaces is equivalent to
Mat and ordinary tensor product of vector spaces. Every monoidal category
is equivalent to a strictly associative one (coherence theorem), and Mat
is a concrete way of doing this for Vect.

Category theorists call your equation "the middle-four-interchange-law"
since it involves interchanging the middle two of a string of four
terms. It is the axiom which makes 2-categories interesting. 

I made much use of all this, including Kronecker product, in my Myhill
Lectures at Buffalo, 20-23 April 1993 [available as Macquarie University 
Math Report 93-130 (June 1993), submitted for publication].

Sincerely,
Ross

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
Date: Tue, 30 Nov 93 14:21:42 EST
From: barr@triples.Math.McGill.CA (Michael Barr)

I'd like to modify it.  I got too clever, forgetting that commuting with
sums doesn't need any additional naturality.

> Date: Mon, 29 Nov 93 09:52:49 EST
> From: stiller@blaze.cs.jhu.edu
>
> Is there a categorical analogue of the following well-known matrix
> identity: If A, B, C, D are matrices over a field and juxtaposition
> denotes matrix multiplication and \otimes denotes Kronecker product,
> then
>
> (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
>
> whenever the dimensions are consistent?
>
> Note: I am a category theory novice, but the proof of this identity
> uses so few of the field axioms, and is so useful, (and curiously
> looks like the interchange of horizontal and vertical compositions of
> natural transformations which I presume is coincidental) that I was
> just curious if an analogue of this identity is true in more general
> categories than matrices over rings.
> --
> Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University.
> stiller@cs.jhu.edu. "Tertan I am, but what is Tertan?  Of this time, of
> that place, of some parentage, what does it matter?"
>
>
>
The stated identity is obviously true in any additive category with
a monoidal structure, provided the monoidal structure commutes with
the finite sums.  This is guaranteed if the monoidal structure is
part of a closed monoidal structure.  Just a consequence of the
functoriality of the finite sums.

Michael



