Subject: categories: Re: query

From: Michael Batanin <[email-masked]>
Date: Sat, 21 Nov 1998 10:06:29 +1100

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.