* strictification
@ 1999-01-06 11:28 James Stasheff
1999-01-06 22:56 ` strictification Michael Batanin
0 siblings, 1 reply; 3+ messages in thread
From: James Stasheff @ 1999-01-06 11:28 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: strictification
1999-01-06 11:28 strictification James Stasheff
@ 1999-01-06 22:56 ` Michael Batanin
0 siblings, 0 replies; 3+ messages in thread
From: Michael Batanin @ 1999-01-06 22:56 UTC (permalink / raw)
To: categories
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.
^ permalink raw reply [flat|nested] 3+ messages in thread
* re: strictification
@ 1999-01-07 7:49 Carlos Simpson
0 siblings, 0 replies; 3+ messages in thread
From: Carlos Simpson @ 1999-01-07 7:49 UTC (permalink / raw)
To: categories
>
> 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
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