Dear David On 27/09/2010, at 12:11 AM, David Leduc wrote:

Let C and D be strict omega-categories. One can build the following strict omega-category that I call [C,D] :

objects = functor from C to D hom_[C,D] (F, G) = Product_{X in C} hom_D (F X, G X)

This construction can be extended to an omega-functor [_,_]. Is [_,_] the internal hom of the omega-category of omega-categories?

The term "the internal hom" is not well defined. There are many monoidal closed structures on omega-Cat. On Cat, there are two useful internal homs that have corresponding tensor products forming symmetric monoidal closed structures. The more familiar is where the internal hom from C to D is the usual functor category [C,D] where objects are functors f : C --> D and morphisms t : f ==> g are families of morphisms t_c : fc --> gc in D, indexed by the objects c of C, which are natural in c. The other internal hom [|C,D|] from C to D gives what I call the funny monoidal structure on Cat: the objects are again functors f : C --> D and morphisms t : f ==> g are families of morphisms t_c : fc --> gc in D, indexed by the objects c of C. No naturality requirement. By the time we get up to omega-Cat there are many monoidal closed structures. The one you describe is one extreme. There is also the one Urs mentioned and called the Crans-Gray structure: it is very important for many purposes. I thought the one you wanted was the cartesian closed structure which I would write as [C,D] for omega-categories C and D. Since we know in this case what the tensor product is, cartesian product, and since mathematicians traditionally like to describe constructions in terms of elements rather than co-elements, there is a straightforward way to discover what an n-cell of [C,D] must be. The n-cells a of any omega- category A are in natural bijection with omega-functors a : 2_n --> A where 2_n is the omega-category with precisely one n-cell and two m- cells for all m < n. So an n-cell of [C,D] must correspond to an omega-functor 2_n --> [C,D] which, if we want cartesian closedness, must correspond to an omega-functor 2_n x C --> D (where I write "x" for cartesian product). So the n-cells of [C,D] can be taken to be omega-functors 2_n x C --> D. It is a moderate exercise to work out how to define the various compositions (the 2_n form a co-omega-category in omega-Cat). Then we need to check that it really is the cartesian internal hom; that is, the set omega-Cat(B,[C,D]) is isomorphic to the set omega-Cat(BxC,D), naturally in B. There's no denying, there's some work but it is all forced. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]