strictification

strictification

[James Stasheff]

Is there a strictification result for A_infty-cats?
If so, under what hypotheses? and by whome? where?

.oooO   Jim Stasheff		jds@math.unc.edu
(UNC)   Math-UNC		(919)-962-9607
 \ (    Chapel Hill NC		FAX:(919)-962-2568
  \*)   27599-3250

        http://www.math.unc.edu/Faculty/jds

Re: strictification

[Michael Batanin]

James Stasheff wrote:
> 
> Is there a strictification result for A_infty-cats?
> If so, under what hypotheses? and by whome? where?
> 
> .oooO   Jim Stasheff            jds@math.unc.edu
> (UNC)   Math-UNC                (919)-962-9607
>  \ (    Chapel Hill NC          FAX:(919)-962-2568
>   \*)   27599-3250
> 
>         http://www.math.unc.edu/Faculty/jds

Yes. Im my paper
"Homotopy coherent category theory and A_{\infty}-structures in monoidal
categories" JPAA, 123 (1988), 67-103, theorems 2.3, 2.4 and corollary
2.3.1.. 

In this paper I define A_{\infty}-categories as algebras in the category
of K-graphs over A_{\infty}-K-operads, where K is a simplicial monoidal
category with Quillen model structure such that tensor commutes 
with simplicial realization functor. I show that every locally fibrant
A_{\infty}-category (i.e. Hom(a,b) is fibrant object in K for every a
and b) is equivalent in some homotopy coherent sense to a honest
K-category.

Michael Batanin.

re: strictification

[Carlos Simpson]

>
> Is there a strictification result for A_infty-cats?
> If so, under what hypotheses? and by whome? where?
>
> .oooO   Jim Stasheff            jds@math.unc.edu

It seems that a reference for this result is a paper of Dwyer-Kan-Smith:

W. Dwyer, D. Kan, J. Smith. Homotopy commutative diagrams and their
realizations. JPAA 57 (1989), 5-24.

This is prior to Batanin's paper (NB there is a typographical error in
Batanin's message---the year of his paper is 1998 not 1988!).

I found D-K-S in my bibliographic wanderings this fall. In the last section
of their paper, they define the notion of ``Segal category'' and at the
same time prove that any Segal category is equivalent to a strict
simplicial category.
The terminology ``Segal category'' is my own (D-K-S don't give this notion
a name). The notion of ``Segal category'' is the Segal-delooping-machine
equivalent of the notion of A_{\infty}-category.

In our preprint of this summer (math.AG/9807049), A. Hirschowitz and I give
a sketch of proof of the strictification result of D-K-S. We were not aware
at the time of D-K-S, nor of Batanin's paper which also gives a proof and
which treats a more general situation too. (I found out about Batanin's
paper this fall thanks to the previous flurry of messages on ``categories''
occasionned by a question from Jim!)

I don't claim to have actually understood DKS's proof because it is very
short and in very abstract language; however, given that (1) all proofs of
this type of strictification result are basically the same; and (2) D-K-S
have a good
track record; there doesn't seem to be any doubt that the proof is indeed
contained in their paper.

The definition of ``Segal category'' in D-K-S is of course much prior to any
of my own versions of this definition. It also seems to be (as far as I know)
the first occurrence of the notion of A_{\infty}-category.

In this context one should point out that Jim's original notion plus all of
the subsequent delooping-machine variants, are just A_{\infty}-categories
with one object; and going to the case of several objects is a rather
obvious embellishment, so discussing ``priority'' for this notion would
seem to be arcane indeed!

---Carlos Simpson