Classifying space

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

$$BC=(⨿{△}_k\times N(C{)}_k)/\sim$$

where the equivalence relation $\sim$ glues the $k$-simplices together 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 $[\mathrm{point}/G]$.