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 Čech (hyper)descent. As an application we study (hyper)sheaves on $(n,1)$-topoi for all $n\in {\mathbf{Z}}_{\ge 1}\cup \{\infty \}$, 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 $n\in {\mathbf{Z}}_{\ge 1}$ 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 $L^{\otimes }:{\mathcal{C}}^{\otimes }\to {\mathcal{D}}^{\otimes }$, and study (hyper)sheafification under a change of universe.