$\infty$-cosheafification

TL;DR

This paper develops the theory of cosheafification within the context of $$-category theory, establishing existence, properties, and equivalences for $$-cosheaves, thus extending duality concepts in higher category theory.

Contribution

It introduces the concept of cosheafification in $$-categories, proves its existence, and shows the $$-category of cosheaves is presentable and equivalent to a category of left adjoint functors.

Findings

Existence of cosheafification in $$-categories.

The $$-category of cosheaves is presentable.

Equivalence between $$-cosheaves and left adjoint functors.

Abstract

Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called \textit{cosheafifications}, in the $\infty$-category theory. We also prove that the $\infty$-category of $\infty$-cosheaves is presentable and equivalent to an $\infty$-category of left adjoint functors.