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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)
##### Keywords:
Category; Classifying space; Sheaf; Representable functor
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