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 on or the other? Even more ridiculously, can we allow that collection to be... I don't know, an $m$-morphism for $m<n$?

Thanks, Jon

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