Internal Homs in Infinity Categories

Question

Given an $\infty$-category $\mathcal{C}$ with whatever descriptors you wish, how do higher internal homs work? Specifically, given two $n$-morphisms, $f$ and $g$, is it possible (I don't see why not a priori) to have the collection of $n+1$-morphisms between $f$ and $g$ itself be an $n$-morphism? Or, is it more sensible to have this collection be an object of the category? Are there any kind of known paradoxes or anything if we allow one or the other? Even more ridiculously, can we allow that collection to be... I don't know, an $m$-morphism for some $m<n$?

Thanks, Jon

Answer 1

I'm sure someone might give a better answer, but here are my two cents. I'd like to start out by pointing out that your question is difficult already for n=0:

First, it would help to know what one means by an enriched oo-category. Other than special, incredibly natural cases like Spaces, Chain complexes, and Spectra, it's hard to know what we mean by this. You should send an e-mail to Rune Haugseng and David Gepner--they're working on the theory of enriched oo-categories.

Second, though I think your question is a cool theoretical question, I don't know of any example of an $\infty$-category where the higher morphisms ($n \geq 2$) form anything but spaces in a reasonable sense. For instance, though spectra and chain complexes are enriched over themselves, 2-morphisms seem most naturally a space, and nothing more. I may just be unaware of some higher enrichment, though. (If so, someone please comment--I'd love to learn.)

So as a result, it seems natural to ask about $(\infty,n)$-categories with various forms of enrichment, rather than just $\infty$-categories. If people are still figuring out enriching $\infty$-categories, I might suspect that enriching $(\infty,n)$-categories is also difficult. A suggestion is to think of examples of higher categories that seem to have a symmetric monoidal structure. If the tensor product seems to want a right adjoint, you might have a chance at understanding internal Homs via some tensor-hom adjunction.

Answer 2

Here is a suggestion that may not be liked much, but has worked well for strict versions of $\infty$-groupoids and categories, namely to use the cubical model.

In

R. Brown, F.A. Al-Agl, R. Steiner, `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71-118.

we showed that strict globular $\omega$-categories are equivalent to strict cubical $\omega$-categories with connections. Now (see section 10) the monoidal closed structure on the latter category is not so hard to write down, following the groupoid version given in 1987 by myself and Higgins.

R. Brown and P.J. Higgins, ``Tensor products and homotopies for $\omega$-groupoids and crossed complexes'', J. Pure Appl. Alg. 47 (1987) 1-33.

Basically this uses the simple idea that for the $n$-cube $I^n$ we have the formula $I^m \times I^n \cong I^{m+n}$.

So the above equivalence of categories allows one to translate the closed monoidal structure in the cubical case to one on strict globular $\omega$-categories. This is related to earlier versions by Street and by Crans.

So the question is: can one use similar cubical methods for weak $\infty$-structures?

The other main advantage of cubical methods for the work with Philip Higgins was the easy description of multiple compositions as arrays of the form $[\alpha_{(r)}]$ where $(r)$ is a multi-index, so allowing algebraic inverses to subdivision.