On $\mathbf {Cat}$-valued sheaves

TL;DR

This paper develops a framework for defining sheaves valued in the 2-category of categories over a topological groupoid, introducing categorical unions and constructing sheaves of local sections, including for categorical groups.

Contribution

It introduces a new approach to ${f Cat}$-valued sheaves over topological groupoids, including the notion of categorical union and the construction of sheaves of local functorial sections.

Findings

Categories of local functorial sections form ${f Cat}$-valued sheaves.

The framework extends to ${f CatGrp}$-valued sheaves for categorical groups.

A new notion of categorical union enables meaningful covers of topological categories.

Abstract

Let ${\widetilde {\mathcal O}}(\mathbf B)$ be the category of (open) subcategories of a topological groupoid ${\mathbf B}.$ This paper concerns with the ${\mathbf {Cat}}$-valued sheaves over category ${\widetilde {\mathcal O}}(\mathbf B).$ Since ${\mathbf {Cat}}$ is not a concrete category, traditional definition of presheaf can not deal with the situation. [13] proposes a new framework for the purpose. Starting from the definition given in [13], we build-up the frame work for ${\mathbf {Cat}}$-valued sheaves. For that purpose we introduce a notion of categorical union, such that categorical union of subcategories is a subcategory, which is required for a meaningful definition of a categorical cover of a topological category. The main result is the following. For a fixed category $\mathbf C,$ the categories of local functorial sections from $\mathbf B$ to $\mathbf C$ define a ${\mathbf {Cat}}$-valued sheaf on ${\widetilde {\mathcal O}}(\mathbf B).$ Replacing $\mathbf C$ with a categorical group $\mathcal G,$ we find a ${\mathbf {CatGrp}}$-valued sheaf on ${\widetilde {\mathcal O}}(\mathbf B).$