Cat-valued sheaves

Let (mathcal{widetilde{O}})(B) be the category of (open) subcategories of a topological groupoid B: In this paper we study Cat-valued sheaves over category (mathcal{widetilde{O}})(B): The paper introduces a notion of categorical union, such that the categorical union of subcategories is a subcategory. We use this definition of categorical unions to define a categorical cover of a topological category. Instead of assuming a Grothendieck topology, we define Cat-valued sheaves in terms of the categorical cover defined in this paper. The main result is the following. For a fixed category C, the categories of local functorial sections from B to C define a Catvalued sheaf on (mathcal{widetilde{O}})(B): Replacing C with a categorical group G; we find a CatGrp-valued sheaf on (mathcal{widetilde{O}})(B): We also relate and distinguish our construction with the notion of stacks.