On $\mathbf{Cat}$-valued sheaves

Abstract

Let $\tilde{\mathcal{O}}(\mathbf{B})$ be the category of (open) subcategories of a topological groupoid $\mathbf{B}.$ This paper concerns with the $\mathbf{Cat}$-valued sheaves over category $\tilde{\mathcal{O}}(\mathbf{B}).$ 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 $\tilde{\mathcal{O}}(\mathbf{B}).$ Replacing $\mathbf{C}$ with a categorical group $\mathcal{G},$ we find a $\mathbf{CatGrp}$-valued sheaf on $\tilde{\mathcal{O}}(\mathbf{B}).$

43 pages, 5 figures; Previous article has been divided into two parts. This is the second part

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