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