What does the classifying space of a category classify? (English) Zbl 1093.57012

Weiss, Michael

Homology Homotopy Appl. 7, No. 1, 185-195 (2005).

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 \(B{\mathcal C}\).

Reviewer: Daniel Tanré (Villeneuve d’ Ascq)

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