# Classifying space

The classifying space of a category is the geometric realization of the nerve . That is,

where the equivalence relation glues the -simplices togetehr as specified by the face and degeneracy maps of . For a group , we can consider the category with a single object and morphisms given by elements of ; in this case, this construction recovers the Borel construction . More generally, given a group acting on a space , we can construct a (topological) category whose objects are given by points in and whose morphisms are given by elements of . The classifying space of this category is the homotopy quotient . If is a strict symmetric monoidal category then will be an infinite loop space.

Note that in algebraic geometry, '' often refers to the stack-theoretic quotient .

Jeffrey Herschel Giansiracusa 2005-06-27