I apologize in advance if this is a rather elementary question; I am a physicist and I am still learning the basics of higher topos theory.
An $(\infty, 1)$-topos $H$ is morally a category of $(\infty, 1)$-sheaves $C^{\text{op}} \to \infty \mathrm{Gpd}$ over an $(\infty, 1)$-site $C$. What I am now interested in are possible generalizations to the setting of $(\infty, n)$-categories. In ordinary $n$-category theory, we have the notion of indexed categories given by $n$-functors $C^{\text{op}} \to (n-1)\mathrm{Cat}$ for $C$ a 1-category. Presumably, there is a generalization to $(\infty, n)$-categories of this construction, from which one could construct an $(\infty, n)$-topos of $(\infty, n)$-sheaves on an $(\infty, 1)$-site $C$.
My question is now this: it seems (as far as I know) that there is only a natural notion of coverage/descent conditions on $(\infty, 1)$-categories that allows the definition of an $(\infty, 1)$-site. However, I could more generally specify an $(\infty, n)$-functor $C \to (\infty, n-1)\mathrm{Cat}$. Is there any way to understand such functors as presheaves? That is, is there a notion of coverage on an $(\infty, n)$-category, if this is even well-defined?
