Subject: categories: Re: query

From: [email-masked] (Ross Street)
Date: Thu, 19 Nov 1998 11:15:46 +1000

>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??

Fortunately, since the A_\infty categories (described sketchily in the
preprint I have of Fukaya) use chain complexes rather than topological
spaces (so that I believe a 1-object A_\infty category is an algebra for
the A_\infty non-permutative operad on chain complexes - an A_\infty
DG-algebra), we do not need to realise the whole homotopy types dream (as
described by John Baez) to give a categorical description of A_\infty
categories. Several years ago I remember discussing this by email with Jim
Stasheff.  Dominic Verity was here at the time.  I cannot remember all the
details but the two basic ingredients were some free strict n-categories
made from the set (whose elements are to be the objects of the A_\infty
category) in much the way the orientals are constructed, and the
construction (below) mentioned in my Oberwolfach Descent Theory notes of
September 1995.

        There is a connection between the Gray tensor product and ordinary
chain complexes.  Each chain complex R gives rise to a (strict)
omega-category J(R) whose 0-cells are 0-cycles  a  in  R, whose 1-cells  b
: a --> a'  are elements  b  in  R_1  with  d(b) = a'- a,  whose 2-cells  c
: b --> b'  are elements  c  in  R_2  with  d(c) = b'- b,  and so on.  All
compositions are addition.  This gives a functor  J :  DG --> omega-Cat
from the category  DG  of chain complexes and chain maps.  In fact,  J :
DG --> omega-Cat  is a monoidal functor where  DG  has the usual tensor
product of chain complexes and  omega-Cat  has the Gray tensor product.  By
applying  J  on homs, we obtain a (2-) functor  J_* :  DG-Cat --> V_2-Cat,
where  V_2  is  omega-Cat  with the Gray-like tensor product (extending the
natural tensor product of oriented cubes as described in Sjoerd Crans
thesis).  In particular, since  DG  is closed, it is a DG-category and we
can apply  J_*  to it.  The V_2-category  J*(DG)  has chain complexes as
0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the
higher cells are higher analogues of chain homotopies.

Best regards,
Ross