Subject: categories: Jim's query

From: "T.Porter" <[email-masked]>
Date: Fri, 20 Nov 1998 14:44:43 +0000

This is not really an adequate reply to Jim's query. The reason is that as
I understand it, he is asking for a source that will explore the homotopy
theory of the `globular' models of weak infty categories. My approach to
the area is in some sense the dual. Starting with the various models for
homotopy types or bits of them, try to see what is mirrored in the weak
infty category theory by the homotopy structure.  This is in some sense the
converse to his query but may none-the-less be relevant.

My intuition was, and still is, that the Kan condition on simplicial sets
gives a composition/pasting up to coherent homotopy. Moreover the `filler'
gives the justification for the composite. (Compare the filler structure of
the nerve of a category with that in an arbitrary Kan complex.) Accepting
that as a starting point, and the idea that the `category' of weak infinity
categories should be a weak infinity category, Jean-Marc Cordier and I
looked at `locally Kan' simplicially enriched categories (e.g. Trans AMS
349(1997)1-54). With that viewpoint, it becomes clear that the structure of
an A_\infty category is needed to make things really `coherent', but that
many of the constructions of `ordinary' category theory have A_\infty or
homotopy coherent analogues in this setting, which thus serves as a
test-bed' for the development of the more general theory.  In part this
relates to Michael Batanin's paper in the Cahiers where explicit
consideration of A_\infty structure is given.

That theory looks at the `global' structure to some extent, but
simplicially enriched groupoids model all homotopy types, so a corollary of
the simplicial to globular type of transition should be that one should be
able to construct weak \infty categroies DIRECTLY from the algebra of a
simplicially enriched groupoid.  The obvious place to look for this is in
the Moore complex which carries a hypercrossed complex structure in the
sense of Pilar Carrasco and Antonio Cegarra.  (This is related to the
n-hypergroupoid structures of Jack Duskin.) Exploring the \infty category
structure, potentially in their definition, is the  aim of another line of
research and in low dimensions, this has been attacked by Ali Mutlu and
myself,  (see very recent articles in TAC or Bangor's preprint list on the
web). 

Getting nearer to Jim's query, any bridge between homotopy theory and
higher dimensional category theory should I feel aim to be approachable by
algebraic topologists and therefore should start with a recognisable model
for homotopy types.

Another approach that must be mentioned is that of Tamsamani and Simpson
using multisimplicial objects. Presumably this also links in with the
cat^n-groupoid approach pioneered some 14 years ago by Loday.  This only
handles n-types but can be extended to a model that has higher information
but in those dimensions above n, the Whitehead products are trivial. (Has
anyone looked at the Whitehead and Samelson products from a globular or
weak \infty category viewpoint?)

The question of simplicial rather than cubical theory is a difficult one.
Marco Grandis made a good case for the cubical formulation the other day,
and the use of Kan filler conditions in a cubical setting has been for a
long time a speciality of Heiner Kamps (see the recent book by him and
me!). That theory does not directly address the question of A_\infty
category structures, but it does raise a question that deserves some
attention. Suppose you want an analogue of a given theorem from homotopy
theory but in another non-topological context and you can get a weak
composition structure in your setting (typically given by fillers for boxes
in a cubical `enrichment'). What fillers/composites do you need for your
particular theorem (e.g.Dold's theorem on homotopy equivalence between
cofibrations, or long Dold-Puppe type sequences)?  

This last question also raises that of the motivation for the
generalisations.  I am convinced these are useful, even important, but
perhaps some debate on directions to explore and goals to seek out might
help in providing a fuller answer to Jim's question.

Tim.

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From: James Stasheff <[email-masked]>
Date: Fri, 20 Nov 1998 17:40:38 -0500 (EST)

Amazing what deep interpretations have been given to my shallow question
though i appreciate the answers.  Al I was after was
some place for a novice reader to see how category theory
and homtopy theory interacted in the concept of A_\infty-category
cf. Smirnov or Fukaya

.oooO   Jim Stasheff		[email-masked]
(UNC)   Math-UNC		(919)-962-9607
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