The language of higher category theory in more than analogous to homotopy theory, especially in the cellular version. What is the appropriate reference for a non-categorical reader? Have the A_\infty categories of Smirnov or Fukaya been treated in the categorical literature??

Fortunately, since the A_\infty categories (described sketchily in the preprint I have of Fukaya) use chain complexes rather than topological spaces (so that I believe a 1-object A_\infty category is an algebra for the A_\infty non-permutative operad on chain complexes - an A_\infty DG-algebra), we do not need to realise the whole homotopy types dream (as described by John Baez) to give a categorical description of A_\infty categories. Several years ago I remember discussing this by email with Jim Stasheff. Dominic Verity was here at the time. I cannot remember all the details but the two basic ingredients were some free strict n-categories made from the set (whose elements are to be the objects of the A_\infty category) in much the way the orientals are constructed, and the construction (below) mentioned in my Oberwolfach Descent Theory notes of September 1995. There is a connection between the Gray tensor product and ordinary chain complexes. Each chain complex R gives rise to a (strict) omega-category J(R) whose 0-cells are 0-cycles a in R, whose 1-cells b : a --> a' are elements b in R_1 with d(b) = a'- a, whose 2-cells c : b --> b' are elements c in R_2 with d(c) = b'- b, and so on. All compositions are addition. This gives a functor J : DG --> omega-Cat from the category DG of chain complexes and chain maps. In fact, J : DG --> omega-Cat is a monoidal functor where DG has the usual tensor product of chain complexes and omega-Cat has the Gray tensor product. By applying J on homs, we obtain a (2-) functor J_* : DG-Cat --> V_2-Cat, where V_2 is omega-Cat with the Gray-like tensor product (extending the natural tensor product of oriented cubes as described in Sjoerd Crans thesis). In particular, since DG is closed, it is a DG-category and we can apply J_* to it. The V_2-category J*(DG) has chain complexes as 0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the higher cells are higher analogues of chain homotopies. Best regards, Ross

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.