Hypersheaves and bases

TL;DR

This paper proves that for any topological space with a basis, the category of hypersheaves valued in an $"infty$-category with limits is equivalent to the subcategory of basic hypersheaves, simplifying their study.

Contribution

It establishes an equivalence between all hypersheaves and basic hypersheaves on a topological space with a basis for any $"infty$-category with limits.

Findings

Equivalence of hypersheaves and basic hypersheaves established.

Simplifies the understanding of hypersheaves on topological spaces.

Applicable to $"infty$-categories with limits.

Abstract

Let $X$ be a topological space equipped with a basis. We prove that, for every $\infty$-category $\mathcal{C}$ with limits, the restriction functor from $\mathcal{C}$-valued hypersheaves on $X$ to basic hypersheaves is an equivalence of $\infty$-categories.