My approach to the area of homotopy coherence and its relations with weak infinity categories is that of starting with the various models for homotopy types or bits of them, then to try to see what is mirrored in the weak infinity category theory by the homotopy structure.

My intuition was, and still is, that the Kan condition on simplicial sets gives a composition/pasting up to coherent homotopy. Moreover the `filler' gives the justification for the composite. (Compare the filler structure of the nerve of a category with that in an arbitrary Kan complex.) This led to studying various formulations of homotopy coherence including that which has since been called `quasi-category theory'. This term has been used by Joyal to refer to what we had called `weak Kan complexes', that is exactly an abstraction of the filler properties of the nerve of a category.

Accepting that as a starting point, and the idea that the `category' of weak infinity categories should be a weak infinity category, Jean-Marc Cordier and I looked at `locally Kan' simplicially enriched categories (e.g. Trans AMS 349(1997)1-54). With that viewpoint, it becomes clear that the structure of an A_\infty category is needed to make things really `coherent', but that many of the constructions of `ordinary' category theory have A_\infty or homotopy coherent analogues in this setting, which thus serves as a test-bed' for the development of the more general theory.  In part this relates to Michael Batanin's paper in the Cahiers where explicit consideration of A_\infty structure is given.

At the same time in paper 50, Jean-Marc and I had shown that the homotopy coherent nerve of simplicially enriched category was a quasi-cateory, if the simplicial enrichement was via Kan complexes.

That theory looks at the `global' structure to some extent, but simplicially enriched groupoids model all homotopy types, so a corollary of the simplicial to globular type of transition should be that one should be able to construct weak \infty categroies DIRECTLY from the algebra of a simplicially enriched groupoid.  The obvious place to look for this is in the Moore complex which carries a hypercrossed complex structure in the sense of Pilar Carrasco and Antonio Cegarra.  (This is related to the n-hypergroupoid structures of Jack Duskin.) Exploring the \infty category structure, potentially in their definition, is the  aim of another line of research and in low dimensions, this has been attacked by Ali Mutlu and myself, (see very recent articles in TAC or Bangor's preprint list on the web).

Any bridge between homotopy theory and higher dimensional category theory should, I feel, aim to be approachable by algebraic topologists and therefore should start with a recognisable model for homotopy types. That way the immense insights given by algebraic topology can be more easily transported to any new area in which higher dimensional category theory  is being applied.

Another approach that must be mentioned is that of Tamsamani and Simpson using multisimplicial objects. Presumably this also links in with the cat^n-groupoid approach pioneered some 14 years ago by Loday.  This only handles n-types but can be extended to a model that has higher information but in those dimensions above n, the Whitehead products are trivial. (Has anyone looked at the Whitehead and Samelson products from a globular or weak \infty category viewpoint?)

The question of simplicial rather than cubical theory is a difficult one. Marco Grandis has made a good case for the cubical formulation, and the use of Kan filler conditions in a cubical setting has been for a long time a speciality of Heiner Kamps (see the recent book by him and me!). That theory does not directly address the question of A_\infty category structures, but it does raise a question that deserves some attention. Suppose you want an analogue of a given theorem from homotopy theory but in another non-topological context and you can get a weak composition structure in your setting (typically given by fillers for boxes in a cubical `enrichment'). What fillers/composites do you need for your particular theorem (e.g.Dold's theorem on homotopy equivalence between cofibrations, or long Dold-Puppe type sequences)?

This last question also raises that of the motivation for the generalisations.  I am convinced these are useful, even important, but perhaps some debate on directions to explore and goals to seek out is needed.