Higher geometric sheaf theories

TL;DR

This paper develops a new framework of higher geometric sheaf theories using higher covering diagrams, unifying descent conditions, and revealing differences from classical sheaf theories through cotopological localizations.

Contribution

It introduces higher covering diagrams and structured colimit pre-topologies, defining a new higher geometric sheaf theory that generalizes and refines existing $ ext{∞}$-categorical sheaf concepts.

Findings

Higher geometric sheaves form a sub-canonical sheaf theory on $ ext{∞}$-categories.

The higher geometric sheaf theory differs from classical infinitary-coherent sheaves via a non-trivial cotopological localization.

The theory characterizes higher $ ext{∞}$-categories and embeds $ ext{∞}$-toposes fully faithfully.

Abstract

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.