Precosheaves of pro-sets and abelian pro-groups are smooth

TL;DR

This paper proves that precosheaves of pro-sets and abelian pro-groups are inherently smooth on topological spaces, establishing a reflector to cosheaves and connecting with shape theory.

Contribution

It introduces a reflector from precosheaves to cosheaves in pro-set and abelian pro-group contexts, showing all precosheaves are locally isomorphic to cosheaves.

Findings

Existence of a reflector from precosheaves to cosheaves.

All precosheaves on topological spaces are smooth.

Connections established with shape theory.

Abstract

Let $\mathbb{D}$ be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site $X$, there exists a reflector from the category of precosheaves on $X$ with values in $\mathbb{D}$ to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.