Subject: cantor-bernstein
Date: Wed, 1 Dec 93 11:36:55 +0100
From: ageron@univ-caen.fr (Pierre Ageron)

The statement and the AC-free proof of Cantor-Schroeder-Bernstein's theorem 
have obviously some categorical content. Has this been already investigated ?
Thank you !
              Pierre AGERON   
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: REMINDER: LICS'94 submission deadline 
Date: Wed, 1 Dec 93 10:26 EST
From: felty@research.att.com (Amy Felty)


                    Ninth Annual IEEE Symposium on
                      LOGIC IN COMPUTER SCIENCE
                    July 4-7, 1994, Paris, France

                SUBMISSION DEADLINE: December 13, 1993

10 hard copies of a detailed abstract (not a full paper) and 20
additional copies of the cover page should be received by December 13,
1993 by the program chair.  This is a FIRM DEADLINE: late submissions
will not be considered.

Program Chair: Samson Abramsky, Attn: LICS, Department of Computing,
Imperial College of Science, Technology and Medicine, 180 Queen's Gate,
London SW7 2BZ, United Kingdom, sa@doc.ic.ac.uk,
Phone: (010-44) 71-589-5111 ext. 5005, Fax: (010-44) 71-581-8024

The full announcement can be obtained by anonymous ftp from
research.att.com, directory /dist/lics, or by emailing
lics-request@research.att.com.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Doctoral Scholarship
Date: Thu, 2 Dec 1993 11:50:58 +1100
From: Barry Jay <cbj@socs.uts.EDU.AU>

                Doctoral Scholarship 

             School of Computing Sciences
           University of Technology, Sydney


The scholarship is linked to the "Shape" project, whose goal is to
design and implement an efficient, typed programming language for
vectors, matrices, and other data structures, based on the separation
of data from shape. The project combines both the development of
theory of shapely types with their incorporation in a programming language.

The project is based at UTS, but has participants in Calgary and
Edinburgh.

Applicants will have completed an undergraduate degree at the Honours
level. They should also have experience in programming language
implementation or semantics, or in category theory.

The stipend is $18,679 (tax exempt). Funding is only guaranteed for 12
months in the first instance. Other possibilities for an immediate
three years of funding also exist. Applicants who are Australian
citizens, or permanent residents who have lived continuously in
Australia for the last twelve months, would not be required to pay
student fees.

For further information, contact Barry Jay (cbj@socs.uts.edu.au). 

Applications should contain a curriculum vitae and the names,
addresses, phone numbers and e-mail of two or three referees from whom
confidential reports can be obtained. They should be sent to:

Dr C. Barry Jay
School of Computing Sciences
University of Technology, Sydney
PO Box 123
Broadway
Australia
2007

Ph: (02) 330-1814

for receipt by 17th December, 1993.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: (A\otimes B)(C\otimes D)=(AC)\otimes (BD)
From:   Richard Wood <rjwood@cs.dal.ca>
Date:   Wed, 1 Dec 1993 12:49:02 -0400

> 
> 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?"
> 
There is of course also the pointwise product of matrices of the same size.
(In the case of complex square matrices I believe this is known as the Schur 
or Schur-Hadamard product.) This product and the usual product of matrices do 
not satisfy the middle four interchange law, in the case of matrices over
a commutative ring. However, there are examples in the study of bicategories
where they do _ in a sense.

Consider the bicategory of categories, profunctors and transformations, PRO_.
For categories X,A the hom category PRO_(X,A) is given by SET_^(A^op x X) and
note that it has sums which, with respect to PRO_, we will call "local" sums.
PRO_ has sums in the usual sense which are as in CAT_. Call them global sums.
Global sum injections have right adjoints and the process of taking right 
adjoints carries global sum diagrams to (global) product diagrams. So an arrow 

in PRO_ from a global sum to a global sum is a matrix of arrows (just as in 
any category in which sums and products coincide). A composite of such 
matrices is 
given by matrix multiplication where the component multiplications are 
composites in PRO_ and the component additions are provided by local sums.

These considerations apply to transformations in PRO_ too so that horizontal
composition of transformations is also given by matrix multiplication when
the objects under consideration are global sums. But vertical composition
of transformations is just pointwise vertical composition.  So here there is a
middle four interchange law between pointwise and ordinary matrix 
multiplication _ but note that the component multiplications are not the 
same. The component multiplications of ordinary matrix product are provided
by horizontal composition _ which middle four interchanges with vertical
composition.

The Kronecker product can also be considered in this example. Global binary
products for CAT_ extend to a tensor product for PRO_. This tensor distributes
over global sum and the remarks of Ross on this question apply to this 
example.

At least this is the story modulo a discussion of the coherent isomorphisms
which are easily spotted. This example admits a number of generalizations and
specializations.

RJ
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: weak topos
Date: Fri, 10 Dec 93 20:43:37 +0100
From: Thomas Streicher <streiche@informatik.uni-muenchen.de>

A suitable notion of model for type theory without extensionality  
principle for predicate seems to be the following :

C  is a weak topos iff  C  is locally cartesian closed and there  
exists a generic mono  g , i.e. any mono of  C  can be obtained as a  
pullback of  g  (that I would call a weak subobject classifier).

Such a category is automatically regular (as one can define  
existential quantification in the internal logic).

Now if one takes  TOP(C) = Map(Split(Rel(C)))  one gets a topos.  
Alternatively one could take the fibration of monos over C thus  
getting a tripos and perform the construction of the associated  
topos.

Now my question are the following :

Under which conditions is C the category of separated objects of  
TOP(C) ?

For which toposes E is Sep(E) a weak topos in the sense of my  
definition ?
We know that the effective topos is an example BUT is it possible for  
non-boolean Grothendieck toposes ?

Thomas Streicher
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject:  Research area of objects and agent behavior
Date: Wed, 15 Dec 93 00:32:49 GMT
From: pinj@di.uminho.pt (Jose Pina Miranda)


Hi,

I'm beginning to research in the area of objects and agent behavior 
(observational properties) and I would like to read something about their
categorical and topological definition. 
I'm also interested in agent hierarchies, behavior inheritance and 
parametrization of agent systems.

Some questions:
  1- Can you send me references of papers that you think I should start 
                reading ?
  2- If possible, can you please send me information about Computer Science or
       Mathematic groups who are working in this area, so that I can learn 
       something with them.
  3- Is there any topological mailing list ?

Thank you,

    Jose Miranda
                   ___      ____  
                     /     /   /   /\  /\
                    /     /---    /  \/  \
                 \-/     /       /        \


Jose E. Pina Miranda        | EMail: pinj@di.uminho.pt
Assistant Lecturer          | X.500: c=PT@o=Universidade do Minho
Departamento de Informatica |            @ou=Departamento de Informatica
Universidade do Minho       |            @cn=  /* not available yet */
Campus de Gualtar           | WWW URL: http://s700.uminho.pt/~pinj
4700 Braga                  | Voice: +351 (53) 617 021         
Portugal                    | Fax: +351 (53) 612 954         

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: EUROFOCS Ad
Date: Thu, 16 Dec 1993 16:22:49 +0000
From: gdp@dcs.ed.ac.uk


                          EUROFOCS
European Institute in the Logical Foundations of Computer Science
                     Fellowship Programme
                     
         (Closing date for applications - 5th January 1994)

The European Institute in the Logical Foundations of Computer Science 
invites applications for PostDoctoral Fellowships. These can be held for 
periods ranging from six months to one year. The goal of the work in the 
Institute is to achieve a unified and applicable theory of the semantics 
and logic of languages used to describe, design and program computing 
systems. The research of the Insititute is characterised by a unified 
approach to foundational problems that emphasises the combination of ideas 
arising in design and practice with ideas originating in logic and allied 
mathematical areas. The applicability of the ideas developed is 
demonstrated by producing experimental systems for specification, 
verification and development. 

The research has both national and EC support, with involvement in several 
ESPRIT Basic Research Actions, including CONFER, CONCUR, COMPASS, 
Categorical Logic in Computer Science and Types for Proofs and Programs. 
Research Topics of Interest include: Semantics of Programming Languages; 
Logic Programming Theory; Object-Oriented Programming; Functional 
Programming Theory; Lambda-Calculus; Formal Development of Programs and 
Systems; Concurrency Theory. 

The Institute sites and contact points are: 

CWI, Amsterdam 
Jaco de Baaker <jaco@cwi.nl> (+31) 20 592 4136

The University of Cambridge;
Andrew Pitts <ap@cl.cam.ac.uk> (+44) 223 334621
 
The University of Edinburgh
Rod Burstall <rb@lfcs.ed.ac.uk> (+44) 31 650 5156

Imperial College, London
Samson Abramsky <sa@doc.ic.ac.uk> (+44) 71 589 5111 x5005

ENS and INRIA, Paris
Giuseppi Longo <longo@dmi.ens.fr> (+33) 14 432 3328

The University of Pisa.
Ugo Montanari <ugo@di.unipi.it> (+39) 50 510 221

INRIA, Sophia Antipolis; 
Gilles Kahn <gilles.kahn@sophia.inria.fr> (+33) 93 65 78 01

APPLICATIONS must be sent to the prospective host site. More general 
information on the Institute or the fellowship programme may be requested 
from Gordon Plotkin <gdp@lfcs.ed.ac.uk> (scientific information), or 
George Cleland <g.cleland@lfcs.ed.ac.uk> (administrative information)

ALL APLICATIONS MUST BE RECEIVED BY THE HOST SITE BY 5TH JANUARY 1994
The initial application should include:

1. Name, Address, e-mail,telephone & fax nos.

2. Academic Qualifications

3. Post-doctoral positions

4. Names, addresses, e-mail and phone numbers of at least two referees

5. Publications

6. A summary of research achievments to date

7. Proposed research programme (3-400 words, written in third person)

8. Expected outcomes of research (<200 words)

9. Reason for choice of host institution (<200 words)

Selection of candidates will be made on a number of factors, including 
quality of research proposed, the potential of the candidate, and the 
contribution of the proposed research to both the host site and the 
Institute. Salary and payment details vary from site to site and should be 
discussed with the host site.

Those candidates selected will be required to work with the host site to 
produce a full fellowship document set which is required by the EC. This 
a non-trivial task for the candidate and will require the selected 
candidate to be available and in communication for the last two weeks of 
January.

Prospective candidates are particularly asked to note the ELIGIBILITY 
CRITERIA for the fellowships. These conditions, imposed by the European 
Commission, are: 

i) The applicant must be a national of one of the Member States of the
European Community or EFTA, or be resident in the European Community.

ii) The applicant must be a national of a country other than that in which
the host institution is established and must not have carried out their
normal activity in that country for more than two years prior to the date
of submission of the application. (EC nationals presently resident in a
non-EC country are eligible providing the proposed fellowship is in a
country other than their country of origin.)

iii) Applicants must be young researchers, without other income, having at
least six years' higher education and who hold a doctoral or equivalent
degree, or, if not, have had two years' research experience following a
post-graduate course.





++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: seeking Michael Wright
Date: Fri, 24 Dec 1993 15:21:45 +0000
From: GARY ALEXANDER <G.R.Alexander@open.ac.uk>

At the suggestion of Martin Hyland we are trying to make contact with Michael
Wr ight, who Hyland thought was living in London. He was in Cleveland, Ohio in
1990/ 91 where he helped Colin McLarty with his book "Elementary Categories,
Elementary Toposes". Colin McLarty, we believe, is active on email.

Can anyone tell us how to contact Michael Wright (or are you on this mailing
list , Michael)? Please reply to Gary Alexander at the Open University, as
g.r.alexander@open.ac.uk.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Revised report on sketches
Date: Fri, 24 Dec 93 15:13:21 -0500
From: cfw2@po.CWRU.Edu (Charles F. Wells)

The final version of the document "Sketches: an outline with
references" is now available by FTP.  The present version
corrects errors in the document distributed in July, 1993,
and contains many new references.

You can use anonymous FTP to get the paper from ftp.cwru.edu. 
The DVI file of the paper is SKETCH.DVI, and the BiBTeX source
of the bibliography is SKETCH.BIB.  The latter is an ASCII file
which requires no knowledge of TeX or BiBTeX to read.  Both are
in the directory math/wells.  The file README.TXT contains
information about other documents in the directory.

I will be glad to send a printed copy of "Sketches: an outline
with references" to anyone who cannot use FTP or DVI files.

I am sending this announcement to two mailing lists; my
apologies if you get it twice.

--
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: Has anybody got a name for this
Date: Tue, 21 Dec 93 10:10:56 GMT
From: <loader@maths.ox.ac.uk>

Let C be a category with finite products, and let F : C -> C be a functor,
along with a natural transformation

F*(X,Y,Z) : hom(X x Y, Z)  -->  hom(X x FY, FZ)

that preserves composition and identities in the appropriate manner. (The
idea is that this gives a behaviour for F on collections of arrows from Y
to Z indexed by X.)

Does anyone have a name for a functor with such a natural transformation?

Note that if C is cartesian closed, then this is the same as having a
natural transformation from Y^Z to (FY)^(FZ) that preserves composition and
identities.

This sort of functor arises in logic as follows: Let A[X] be a formula with
a positively occuring propositional variable X.  Then 

   B |- C
------------
A[B] |- A[C]

is a derived rule, making A[-] into a functor.  But we can also do this
with additional side formulae:

   D, B |- C
---------------
D, A[B] |- A[C]

giving a natural transformation as described as above.

I'd use the term "internal functor", except that this is already used for a
functor between two internal categories.

Ralph Loader.

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Subject: Re: Has anybody got a name for this(several posts)
From:   Richard Wood <rjwood@cs.dal.ca>
Date:   Fri, 24 Dec 1993 10:27:32 -0400

If V_ is a symmetric monoidal category but not closed there is stilll a
sense in which V_ itself is a V_-category. After all, V_ "acts" on itself
via tensor:V_xV_--->V_ and it is useful to think of an arrow
X ten Y --->Z as providing an X-indexed family of arrows from Y to Z.

If A_ and B_ are categories equipped with actions of V_ one can ask for
the relevant notion of functor from A_ to B_ that respects such actions
and the condition is the one that you enquire about, your specific case
involving endofunctors on the base which having binary products can be
seen as a symmetric monoidal category.

If the idea of A_ having families of arrows indexed by the objects of V_
is taken as primitive we are led to consider categories A_ equipped
with a functor A_^op x V_^op x A_ ---> SET that is suitably compatible
with the ordinary hom functor for A_. The relevant conditions ultimately
ensure that A_ has the structure of a SET^(V_^op) category. This last is
considered to have the monoidal ((bi)-closed) structure introduced by 
Brian Day.

If the data for such is presented as I have it in the last paragraph then
the idea is that for A and B in A_ and X in V_ the value of the structure
functor at (A,X,B) provides a SET of X-indexed arrows from A to B. This
structure admits the possibility of representability in each of the 3
variables so that SET^(V_^op)-category includes the notions of V_-category,
"V_-tensored category" and "V_-cotensored category".

If A_ and B_ are SET^(V_^op) categories then the notion of SET^(V_^op)
functor F between them is clear but there are 3x3=9 special cases that
bear inspection in case  both A_ and B_ are any of the special cases mentioned
above. In each case "the effect of F on homs" (I believe that is the term 
you want) admits a compact description. In my 1976 thesis there is a table
which displays them.

Best regards
RJ
+++++++++++++++++++++++++++++++++++++++++
Date: Fri, 24 Dec 1993 13:26:43 -0500 (EST)
From: MTHFWL@ubvms.cc.buffalo.edu


<< 
Let C be a category with finite products, and let F : C -> C be a functor,
along with a natural transformation

F*(X,Y,Z) : hom(X x Y, Z)  -->  hom(X x FY, FZ)

that preserves composition and identities in the appropriate manner. (The
idea is that this gives a behaviour for F on collections of arrows from Y
to Z indexed by X.)  >>

It seems that Ralph Loader has rediscovered an important case of what are
called Enriched Functors, see eg Max Kelly's book.
He also notes that these are distinct from Internal Functors. Internal
and Enriched are both included under and interact within the "indexed"
(I would prefer "parameterized") theory treated in another important book:
Springer LNM 661

++++++++++++++++++++++++++++++
Date: Mon, 27 Dec 93 19:35 EST
From: Fred E J Linton <0004142427@mcimail.com>

Data of the form  F*(X,Y,Z): hom(XxY, Z) --> hom(XxFY, FZ)  appear in the
notion of a multilinear functor.  See my Multilinear Yoneda Lemmas article
in an ancient Springer Lecture Notes in Mathematics.    -- Fred

+++++++++++++++++++++++++++++
Date: Tue, 28 Dec 1993 11:21:38 +0100
From: kock@mi.aau.dk


The extra structure on the endofunctor F given by 
   hom(X x Y,Z) ---> hom(X x FY,FZ)
is called a tensorial strength, for reasons deriving from 
enriched category theory, cf Kock: Strong Functors and 
Monoidal Monads, Archiv der Math. (Basel) 23 (1972), 113-120. 
In loc.cit., it is given in terms of
    X x FY ---> F(X x Y),
but such data is equivalent to data of the above kind, by a 
Yoneda-like argument (put Z = X x Y, and apply to the identity 
map of X x Y).

A kind of structure that implies such monoidal strength is the 
data of F being an indexed endofunctor (assuming the category 
in question has pull-backs, thus being indexed over itself). 
This seems to be the case for any endofunctor F on a topos, 
provided F is described by logical means.

Anders Kock     

+++++++++++++++++++++++++++++++
From: dyetter@math.ksu.edu (David Yetter)
Date: Tue, 28 Dec 93 9:44:09 CST

I have not seen a name for the structure R. Loader inquires about. It may be
a lax version of a structure which has been considered in the context of
"quantum topology" (cf. my paper in Topology '90):

        Let C be a monoidal category (e.g. category with (chosen) products),
let X be category equipped with an "action" of C, i.e. a functor
a:C x X ----> X, satisfying the "obvious" coherence conditions. X is then
a "C-module" and a functor F:X--->X such that for all c in C, x in X
such that there is a natural isomorphism from F(a(c,x)) to a(x,F(x))
again satisfying "obvious" coherence conditions is a "C-module functor".

        In the case in question C = X, tensor = a = cartesian product.
I think, however, that one wants a natural transformation, not a 
natural isomorphism.

Best Thoughts,
David Yetter

