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