Higher geometric sheaf theories

We introduce the notion of a higher covering diagram in a base $\infty$-category $\mathcal{C}$. The theory of higher covering diagrams in $\mathcal{C}$ will be shown to recover various descent conditions known from the $\infty$-categorical literature in a uniform manner. In fact, higher covering diagrams always assemble to what we refer to as a structured colimit pre-topology on the base $\mathcal{C}$. It hence always defines a sub-canonical sheaf theory over $\mathcal{C}$, and indeed defines the canonical such whenever $\mathcal{C}$ has pullbacks. This ``higher geometric'' sheaf theory will be shown to differ from the usual infinitary-coherent sheaf theory by a cotopological localization whenever $\mathcal{C}$ is infinitary-coherent itself. We prove that this localization is generally non-trivial. For instance, every $\infty$-topos is the theory of higher geometric sheaves over itself, but the according infinitary-coherent sheaf theory over it is generally strictly larger. The higher geometric sheaves are hence characterized by a limit preservation property that is generally not captured by the classical sheaf condition. We define an $\infty$-category of higher geometric $\infty$-categories, and show that the (opposite of the) $\infty$-category of $\infty$-toposes embeds fully faithfully therein. We show that the higher $\kappa$-geometric sheaf theory on a higher $\kappa$-geometric $\infty$-category defines the free $\infty$-topos generated by it, and consequently that it faithfully generalizes Lurie's definition of a ``sheaf'' over an $\infty$-topos.