From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/947 Path: news.gmane.org!not-for-mail From: Michael Batanin Newsgroups: gmane.science.mathematics.categories Subject: Re: query Date: Sat, 21 Nov 1998 10:06:29 +1100 Organization: Maquarie University, Sydney Message-ID: <3655F5F5.4C15@mpce.mq.edu.au> References: <199811190226.NAA02248@macadam.mpce.mq.edu.au> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=koi8-r Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017363 28436 80.91.229.2 (29 Apr 2009 15:02:43 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:02:43 +0000 (UTC) To: categories@mta.ca Original-X-From: cat-dist Sun Nov 22 01:05:22 1998 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id AAA13274 for categories-list; Sun, 22 Nov 1998 00:04:40 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 3.01Gold (Win95; I) Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 76 Xref: news.gmane.org gmane.science.mathematics.categories:947 Archived-At: This is a summary of my correspondence with J.Stasheff. James Stasheff wrote: > As monoids can be described as categories with one object, > one can consider \Aoo structures on categories with a notion of > homotopy, e.g. topological categories or differential graded categories. > To be more precise, the set of objects and the set of morphisms > carry a notion of homotopy. As usual, one deals with composable > morphisms and then weakens the axiom of associativity up to homotopy > in the strong sense in order to > define \Aoo - categories. This was first done by Smirnov by 1987 > \cite{smirnov:baku} to handle functorial homology operations and > their dependence on choices (cf. indeterminacy). More recently, Fukaya > \cite{fukaya:1} reinvented \Aoo -categories with remarkable applications to > Morse theory and Floer homology. > Inspired by this work, Nest and Tsygan have proposed an \Aoo -category > with automorphisms of an associative algebra as objects and for > the space of morphisms, a twisted version of the Hochschild complex > of the corresponding endomorphism algebras. Michael Batanin: One can generalize "ordinary" category theory in the different ways. One can consider internal category theory, enriched category theory. We can also consider a category as a special sort of simplicial set. All this points of view have their own A_{\infty}-analogues. I realize, that the approach of Smirnov, Fukaya and others is a generalization of "internal" category theory. In my paper "Monoidal globular categories as a natural environment ..."(Adv.Math. 136, 39-103 (1998)) I also consider a Cat-internal version of A_{\infty}-\omega-category (so it involves a weak form of interchange law)that I call monoidal globular category. A surprising coherence theorem sais that a general monoidal globular category is equivalent to a strict one (the internal category structure on objects aloows to strictify interchange low). In another my paper "Homotopy coherent category theory and A_{\infty}-structures in monoidal categories" (JPAA 123(1998),67-103) I defined an enriched version of A_{\infty}-category. So we have a honnest set of objects but morphisms are objects of a monoidal simplicial categories with a Quillen model structure. I also can define what A_{\infty} functor is and prove an appropriate coherence and homotopy invariance theorems. Another nice theorem sais that A_{\infty}-categories and their A_{-infty}-functors form an A_{infty}-category in a natural way. The simpliocial point of view on A_{\infty}-categories goes back to Boardman and Vogt book. The corresponding notion is a simplicial set satisfying some weak Kan conditions. This approach was extensively used by T.Porter and J.-M.Cordier (see T.Porter's answer on Jim's query). James Stasheff: > Since in a category we are concerned with $n$-tuples of morphisms > only when they are composable, it is appropriate to similarly relax > the composition operations for in defining an operad. the result is > known as a partial operad and appears in two different contexts: > in the mathematical physics of vertex operator algebras (VOAs) > \cite{yizhi} and Mivhael Batanin: In my work "Globular monoidal categories ..." I introduced the n-dimensional operads over trees. A 1-dimensional operad in this sense is not exatly the same as usual non-symmetric operad as every operation may have source and target and we can multiply just composable chains of operations. A one object version of this may be identify with a usual nonsymmetric operad. (In my paper I use \omega-operads). I wonder if a partial operad is the same as my 1-operad? Michael.