Classifying space

The classifying space $ BC$ of a category $ C$ is the geometric realization of the nerve $ N(C)$. That is,

$\displaystyle BC = (\amalg \bigtriangleup_{k} \times N(C)_{k})/\sim$

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

Note that in algebraic geometry, ``$ BG$'' often refers to the stack-theoretic quotient $ [\mathrm{point} / G]$.

Jeffrey Herschel Giansiracusa 2005-06-27