Geometric Categories and Sheaves on Topoi

TL;DR

This paper introduces geometric ategories and studies (hyper)sheaves on them, characterizing descent conditions and relating sheaves on ategories to those on ategories of lower truncation levels, with applications to topoi.

Contribution

It defines geometric ategories, characterizes sheaves via ch descent, and links sheaves on ategories to truncated sheaves on lower-level topoi, expanding the theory of sheaves in higher category theory.

Findings

Sheaves on geometric ategories are characterized by ch (hyper)descent.

The effective epimorphism topology on an ategory topoi is the canonical topology.

Studying sheaves on ategories reduces to studying truncated sheaves on lower ategories.

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.