Cosheafification

TL;DR

This paper proves the existence of a coreflection called cosheafification for precosheaves on Grothendieck sites, providing explicit constructions and connections to shape theory, especially when values are in pro-objects.

Contribution

It introduces a cosheafification process for precosheaves valued in locally presentable categories, with explicit constructions for pro-objects and applications to topological spaces.

Findings

Existence of cosheafification for precosheaves in certain categories.

Any precosheaf with values in pro-objects is smooth and locally isomorphic to a cosheaf.

Connections established between cosheaves and shape theory.

Abstract

It is proved that for any Grothendieck site $X$, there exists a coreflection (called $\mathbf{cosheafification}$) from the category of precosheaves on $X$ with values in a category $\mathbf{K}$, to the full subcategory of cosheaves, provided either $\mathbf{K}$ or $\mathbf{K}^{op}$ is locally presentable. If $\mathbf{K}$ is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category $\mathbf{Pro}% \left( \mathbf{K}\right) $ of pro-objects in $\mathbf{K}$. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in $\mathbf{Pro}\left( \mathbf{K}\right) $ is $\mathbf{smooth}$, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.