201. Cat-valued sheaves
- Author
-
Saikat Chatterjee
- Subjects
Subcategory ,Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Topological category ,Grothendieck topology ,Cover (topology) ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,0103 physical sciences ,Sheaf ,010307 mathematical physics ,0101 mathematics ,Categorical variable ,Mathematics - Abstract
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.
- Published
- 2018