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



Internal Homs in Infinity Categories
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.


