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From: Michael Batanin <mbatanin@mpce.mq.edu.au>
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Subject: Re: strictification
Date: Thu, 07 Jan 1999 09:56:57 +1100
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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.