Dear Ross, concerning the internal hom of strict omega-categories:
Given two strict omega-categories C and D, how do you define the strict omega-category of omega-functors between C and D?
There is the Crans-Gray tensor product on StrOmegaCat that makes it biclosed monoidal.
I think David was asking about the simpler cartesian closed structure on omega-Cat. This is constructed in
The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335
You know all this, but for the record I say the following: The cartesian closed structure has an internal hom that is a restriction of the internal hom wrt the Gray structure. Usually the one of the Gray structure is the one of interest. It is the one closer to the full oo-category theoretic notion (the one with no strictness constraints whatsoever). The Crans-Gray tensor product with its property that G^k otimes G^l is k+l-dimensional is the fix in the globular model for what in the simplicial model is automatic, namely that Delta^k x Delta^l is k+l-dimensional. That this is automatic for the cartesian product in simplicial sets but requires more work for globular sets is one of the reasons why simplicial models for oo-categrories are more highly-developed than globular ones: they are easier. A good brief introduction to this is on the first few pages of Sjoerd Crans' A tensor product for Gray-categories http://www.emis.de/journals/TAC/volumes/1999/n2/5-02abs.html (After that introduction the article goes on to refine the globular Gray tensor product to the case of _weak_ (or rather: semi-strict) 3-categories.) But of course for the purposes of David's application (which I don't know about) the strict version of the internal hom might be sufficient. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]