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