Geometric Categories and Sheaves on Topoi

Abstract

We introduce the notion of a geometric $(\infty,1)$-category, the protopyical example of which is an $(\infty,1)$-topos. We study (hyper)sheaves on geometric $(\infty,1)$-categories, proving that these are characterized by a form of \v{C}ech (hyper)descent. As an application we study (hyper)sheaves on $(n,1)$-topoi for all $n\in \mathbf{Z}_{\geq 1}\cup \{\infty\}$, and prove that the effective epimorphism topology on an $(n,1)$-topos $\mathcal{X}$ may be identified as the canonical topology on $\mathcal{X}$. Moreover, we show that for finite $n\in \mathbf{Z}_{\geq 1}$ the study of sheaves on an $(n,1)$-topos $\mathcal{X}$ is equivalent to the study of $(n-1)$-truncated sheaves on certain $(\infty,1)$-topoi. We then globalize our study to consider sheaves on $\infty\mathcal{T} op$. In the appendix, we study the behavior of modules under a reflective monoidal $(\infty,1)$-functor $L^\otimes:\mathcal{C}^{\otimes}\rightarrow \mathcal{D}^{\otimes}$, and study (hyper)sheafification under a change of universe.