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I'm working with globular operadic higher categories (with the Batanin/Leinster definitions) and ending up working a lot in the categories $P$-$\mathrm{Alg}$ of algebras for some globular operad $P$, together with (literal, strict, algebraic) morphisms between them. A familiar case of these is $\textrm{Bicat}_\textit{str}$: the category of bicategories and strict functors between them.

It would be very handy if there were some kind of mapping space constructions in these categories --- that is, some reasonable monoidal closed structures on them. Does anyone know what's been shown, either to exist or not to, either in the general case, or (more likely) for $\textrm{Bicat}_\textit{str}$?

(The most obvious specific candidate, of course, is Cartesian closure. However, even $\textrm{Bicat}_\textit{str}$ fails to be Cartesian closed: chasing through the Yoneda argument that shows what a Cartesian closure would have to look like if it did exist leads one to a counterexample showing that the product doesn't preserve pushouts. The next best hope would presumably be some sort of Gray tensor product; this is where I've not yet been able to find anything further on the closure question.)

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You may know this, but there are people thinking about this sort of question at least from the monoidal viewpoint. For instance, there is this paper, which shows that if you can defined what you want to mean by "a category enriched in P-Alg," then you can automatically recover from that a corresponding monoidal structure on P-Alg. In particular, this can produce the Gray tensor product from the notion of Gray-category. Although in general what you get is only a lax monoidal structure, and I don't think they have (yet) asked when it will be closed.

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Thanks — yes, I have looked at that (though not in detail), but as you say it doesn't (iirc) start looking into the question of closure; and unfortunately, the mapping spaces are what I really want... –  Peter LeFanu Lumsdaine Apr 29 '10 at 17:31

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