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.