Re: query

[john baez <baez@math.ucr.edu>]

Jim Stasheff writes:

> The language of higher category theory in more than analogous
> to homotopy theory, especially in the cellular version.  What
> is the appropriate reference for a non-categorical reader?
> Have the A_\infty categories of Smirnov or Fukaya been treated
> in the categorical literature??

Unfortunately the truly appropriate reference has not yet been
written, because the equivalence between weak infinity-groupoids
and homotopy types has not yet worked out in full detail, at least
not in the cellular version.   The *dream* of translating all of 
homotopy theory into higher category theory is outlined in:

John Baez and James Dolan, Categorification, to appear in Proceedings
Workshop on Higher Category Theory and Mathematical Physics at 
Northwestern University, Evanston, Illinois, March 1997, eds. Ezra
Getzler and Mikhail Kapranov, preprint available as math.QA/9802029.

This also has lots of references to different places where various
bits of the dream have been realized.  

Basically the dream consists of working out the following correspondence:

HIGHER CATEGORY THEORY             HOMOTOPY THEORY

omega-groupoids                    homotopy types
n-groupoids                        homotopy n-types
k-tuply groupal omega-groupoids    homotopy types of k-fold loop spaces
k-tuply groupal n-groupoids        homotopy n-types of k-fold loop spaces
k-tuply monoidal omega-groupoids   homotopy types of E_k spaces
k-tuply monoidal n-groupoids       homotopy n-types of E_k spaces
stable omega-groupoids             homotopy types of infinite loop spaces
stable n-groupoids                 homotopy n-types of infinite loop spaces
Z-groupoids                        homotopy types of spectra
 
How do A_infinity categories fit in?  As far as I can tell they should
correspond to omega-categories where all j-morphisms are invertible for
j > 1.