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What does the classifying space of a category classify? (English) Zbl 1093.57012
Let \({\mathcal C}\) be a small category. A \({\mathcal C}\)-set is a contravariant functor from \({\mathcal C}\) to the category of sets. A \({\mathcal C}\)-set is called representable if it is isomorphic to \(b\mapsto \text{ mor}_{\mathcal C}(b,c)\) for fixed \(c\in {\mathcal C}\). In the paper under review, the author proves that the classifying space \(B{\mathcal C}\) classifies sheaves of \({\mathcal C}\)-sets with representable stalks. The proof uses the construction of a canonical sheaf of \({\mathcal C}\)-sets on \(B{\mathcal C}\).

MSC:
57T30 Bar and cobar constructions
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18G55 Nonabelian homotopical algebra (MSC2010)
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