Date: Tue, 9 Jul 91 11:41:38 GMT-0400
From: jds@rademacher.math.upenn.edu

The following `must have' been considered already.
It is possible to define homotopy groups for a simplicial set.
How about for an n-category? or anyway a 2-category?
Naively, the problem seems to be how to `add' 2-cells, but
now that we have good pasting theorems....

================================
Date: Tue, 9 Jul 91 10:01:50 EDT

                   Seventh Annual IEEE Symposium on
                      LOGIC IN COMPUTER SCIENCE
              June 22--25, 1992, Santa Cruz, California

               PRELIMINARY CALL FOR PAPERS

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PROGRAM CHAIR:
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University of Pennsylvania
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lics@cs.cmu.edu
==============================
From: mas010@vaxa.bangor.ac.uk
Date: 11 July, 1991

In the groupoid case, there are various possibilities for `higher
dimensional objects', namely crossed complexes, infinity-groupoids,
omega-groupoids, simplicial T-complexes, poly-T-complexes (see the references
in my article on `Some problems in non-abelian homological and homotopical
algebra' SLNM 1418, ed M Mimura).

Crossed complexes are useful because of their close relations to classical
tools (chain complexes, universal covering spaces, relative homotopy groups).

Omega-groupoids (=cubical infinity-groupoids with connections, and so
thin structures) are useful because of the clear compositions, monoidal
closed structures, and context for proving the generalised Van Kampen Theorem.

Simplicial T-complexes are useful because of the relation with classical
simplicial techniques, and because of a lingering suspicion in some quarters
that it is old fashioned to use cubical sets.

The aim of the poly-theory was to to give a context for group theory methods
such as Van Kampen diagrams (see D.Johnson, Presentations of groups, LMS
Lecture notes). In any case, what is wrong with pentagons, or rhombic
dodecahedra? How do Stasheff polyhedra fit into a general theory similar to
simplicial sets? What algebraic structures do Stasheff T-complexes
correspond to?

What is not so clear is what are the valuable features of infinity-groupoids.
It is somehow interesting to know they are there, but what does one do with
them when one has got them? I wish I knew. For example, translating the
monoidal closed structure on omega-groupoids to the infinity-groupoid case,
with explicit formulae, seems a formidable task. (It is not so good for
the simplicial case, either!)
There are also problems. For example, what is the crossed complex corresponding
to the free infinity-groupoid on one generator of dimension n? It should be
the fundamental crossed complex of the n-cell with its globular (hemi-spherical
) subdivision. But how does one prove this? The corresponding results for
the cubical and simplicial case use basic facts on the homotopy theory of
geometric realisations for the (current) proofs.

A paper of Spencer and Wong (CTGD 24 (1983) 164-192) shows the advantage
of working with double categories with thin structure rather than with
2-categories. Part of the aim of the thesis of F. Al-Agl was to get similar
methods in the n-dimensional case. Intuitively, one hopes to replace
pasting arguments by calculations with thin elements, since these obey
the transparent rule: any composition of thin elements is thin. One can
also make inductive arguments about how degenerate is a thin element
(see Proposition 4.4 of Brown-Higgins JPAA 22 (1981) 11-41: the whole theory
of omega-groupoids is designed to make this Proposition work, since the
immediately following
lemma gives the crucial result that certain choices previously made do not
affect the final composition, and this is proved by showing that an (n+1)-
dimensional cube is degenerate in direction n+1, so that its two opposite
faces in this
direction are the same). Trying to do this last proof of the Van Kampen
Theorem in infinity-groupoids is a nightmare of intricately coiling tubes
(see Whitney's tube systems?), so that the work with Loday on cat-n-groups
adopted an entirely different approach, using sophisticated simplicial
techniques, which in some mysterious way accommodate all these local problems.

Clearly n-categories arise in nature. But it would seem useful to assess
the advantages and disadvantages of the various possible categories in which
to work before committing oneself to setting up a homotopy theory in one
setting rather than another. The advantage of having these (highly non-trivial)
equivalences of categories is that one can swap round at will, not noticing
the technical nature of the machine which allows one to do this. But as
Vaughan Pratt emphasised, there is still the problem of proving that
infinity-categories are equivalent to omega-categories (=cubical
infinity-categories with connections), and this is not yet solved by the work
of Al-Agl/Steiner, although that does give a beautiful cubical setting
equivalent to infinity-categories. I was very interested to see the high
priority given by Vaughan to this question.

Ross Street in an extra informal lecture at Montreal explained the background
in constructing `cohomology with coefficients in an n-category' for his
work on nerves of infinity-categories (following up suggestions of John
Roberts). I have tended to concentrate on the corresponding possibilities
for the groupoid case, but work of Pachkoria explained at Montreal shows
the geometric possibilities of cohomology with coefficients in a commutative
monoid. Recent work of Larry Breen gives Schreier systems with coefficients
in a crossed square, while Bullejos and Cegarra have considered coefficients
in a reduced `crossed module of length 2' (a la Conduche). Brown-Higgins deal
in a recent preprint (`The classifying space of a crossed complex', MPCamb
Phil Soc, in press) with coefficients in a crossed complex. It all looks
very promising.

Ronnie Brown
===================================
Date: Thu, 11 Jul 91 14:17:03 -0400
From: stark@sbcs.sunysb.edu (Eugene Stark)

Carboni and Walters (Cartesian bicategories I, JPAA 49(1987), p 13) state:

 "Consider a bicategory B with a tensor product.  Then the tensor
 product is the biproduct iff every object has a cocommutative
 comonoid structure and every arrow is a comonoid homomorphism."

Presumably this fact should specialize to the case of symmetric strict monoidal
categories, so that for a symmetric strict monoidal category C, the tensor
product is the cartesian product iff every object has a cocommutative comonoid
structure and every arrow is a comonoid homomorphism.

Let C be a symmetric strict monoidal category, with identity object I,
tensor product denoted by *, and with symmetry isomorphisms sXY: X*Y --> Y*X.
Suppose each object X in C has a cocommutative comonoid structure given
by the arrows:

 tX: X --> I   dX: X --> X*X

The comonoid equations are (dropping the names of the objects for simplicity):

 (1*t)d = 1   (counit)
 (t*1)d = 1
 (d*1)d = (1*d)d   (coassociativity)
 sd = d    (cocommutativity)

Suppose further that every arrow of C is a comonoid homomorphism.
It should then follow that the tensor product is the cartesian product.
Presumably the projections would be (1*t): X*Y-->X and (t*1): X*Y-->Y,
and the target tupling of f: Z-->X and g: Z-->Y would be given by (f*g)d.
The equations that would have to be satisfied are:

 (1X*tY)(f*g)dZ = f  (tX*1Y)(f*g)dZ = g

  (((1X*tY)h)*((tX*1Y)h))dZ = h

The first two are easy to prove.  Can someone tell me how to prove the third?
I am probably missing something simple, but I spent a few hours on it and I
just don't see it.  I can reduce it to the problem of establishing either

  ((1X*tY)*(tX*1Y))d = 1

or else

  d(X*Y) = (1X * sXY * 1Y)(dX * dY)

but I don't see how to prove either of these from the stated assumptions.

       - Gene Stark

==================================
Date: 11 Jul 91 14:03:34 PDT (Thu)
From: pratt@cs.stanford.edu


As I'm sure someone else will say in more detail than I'm qualified to,
your question is currently of considerable interest to n-category
theorists (Ross Street and Mike Johnson in particular), topologists
(Ronnie Brown, Richard Steiner, F. Al-Agl), and at least one computer
scientist, namely myself in connection with modeling concurrent
computation.

A (the?) central question is whether omega-categories (Brown's
terminology) (based on either simplexes or cubes) are equivalent to
infinity-categories (based on n-cells) or just a special case.

For myself, I'm very interested in applying homotopy *monoids* to
improving on our the existing notion of computation as a path,
generalizing it to a homotopy class.  My paper "Modeling Concurrency
with Geometry" (available by anonymous ftp as pub/cg.{tex,dvi} from
boole.stanford.edu, see pub/README for abstracts of related work)
defines a concurrent automaton with n concurrent processes to be an
n-category, and (elsewhere in the paper) suggests the notion of
monoidal homotopy without giving a knock-down definition of it, and
contrasts it with group homotopy.

Since then Rob van Glabbeek has encouraged me to look at neither
simplicial sets nor n-categories but rather cubical sets.  For now
these seem somewhat more tractable than n-categories for modeling
concurrency, and better capture the essence of concurrent automata.
n-categories seem just a bit too general.  However I'm still quite
open-minded about this since I don't see really persuasive arguments
for either.

Vaughan Pratt

=========================
Subj:	T-Algebras in PLC
Date: Thu, 11 Jul 91 14:46:56 +0200
From: Martin Hofmann <mnhofman%faui77.informatik.uni-erlangen.de@unido.bitnet>


During the lecture of Bainbridge,Freyd Scedrov,Scott's-paper

"Functorial Polymorphism" (TCS 90) the following questions arose:


\item Does the authors' construction of an interpretation of PLC d
efinitely imply that there is
a categorical PLC-model in the sense of Reynolds in which

$\forall \alpha.(T(\alpha)->\alpha)->\alpha$

is interpreted as the initial T-Algebra?

\item Does this model (if there is one) have an object with
two distinguishable points, or equivalently is true <> false ?

\item Does the model construction described extend to a model
of the Calculus of Constructions or even ECC, so that we could
consistently add an axiom claiming the initiality of
$\forall \alpha.(T(\alpha)->\alpha)->\alpha$ using Leibniz's
equality (And  $not (true = false)$ of course)?

Could anyone -- perhaps the authors themselves -- help
me to get answers to these problems?

-- Martin Hofmann

P.S. I'm a grad-student working on verification of
ML-programs with ECC and I've represented ML's datatypes
using the above quoted T-algebra-expression.

====================================
Subj:	Query on Topos of Labeled Trees
Date: Thu, 18 Jul 91 12:05:30 -0700
From: Purandar Bhaduri <pbhaduri@yoda.eecs.wsu.edu>

I am a graduate student working in the area of categorical models
of process behavior. One of the models I am looking at is the
topos of labeled trees (cf. Enriched Categorical Semantics for
Distributed Calculi, S.Kasangian and A.Labella, to appear in Jn.
of Pure and Algebra). In this paper and elsewhere there are
references to unpublished manuscripts of Benabou, which are difficult
to get hold of. My question is regarding the subobject classifier
in this topos of labeled trees : What does it look like, for a given
set of labels A? Also, are there any published works on the topos
of labeled trees where its properties are spelled out in detail?
Any help will be greatly appreciated.
Sincerely,
Purandar Bhaduri.

===================================
Subj:	New email address
Date: Mon, 29 Jul 91 15:22:16 +1000
From: street@macadam.mpce.mq.edu.au (Ross Street)


Dear Bob,

The problem with my email has been resolved. The old address will still
work but is being phased out.

Please advise your bulletin board subscribers that my new address is:

      <street@macadam.mpce.mq.edu.au>

Best regards,
Ross
