Editing Classifying space (section)

{{Short description|Quotient of a weakly contractible space by a free action}}
In [[mathematics]], specifically in [[homotopy theory]], a '''classifying space''' ''BG'' of a [[topological group]] ''G'' is the quotient of a [[weakly contractible]] space ''EG'' (i.e., a topological space all of whose [[homotopy group]]s are trivial) by a proper [[free action]] of ''G''. It has the property that any ''G'' [[principal bundle]] over a [[paracompact]] manifold is isomorphic to a [[pullback bundle|pullback]] of the principal bundle <math>EG \to BG</math>.<ref>{{Citation | last1=Stasheff | first1=James D.|author-link=Jim Stasheff | title=Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) | publisher=[[American Mathematical Society]] | year=1971 | chapter=''H''-spaces and classifying spaces: foundations and recent developments | pages= 247–272 Theorem 2 |url=http://www.ams.org/books/pspum/022/ |mr=0321079 |chapter-url={{GBurl|p-wCCAAAQBAJ|p=247}} |doi=10.1090/pspum/022/0321079 |isbn=978-0-8218-9308-1
}}</ref> As explained later, this means that classifying spaces [[representable functor|represent]] a set-valued [[functor]] on the [[homotopy category]] of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of [[topological space]]s, such as [[Sierpiński space]]. This notion is generalized by the notion of [[classifying topos]]. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. 

For a [[discrete group]] ''G'', ''BG'' is a [[connected space|path-connected]] [[topological space]] ''X'' such that the [[fundamental group]] of ''X'' is isomorphic to ''G'' and the higher [[homotopy groups]] of ''X'' are [[trivial group|trivial]]; that is, ''BG'' is an [[Eilenberg–MacLane space]], specifically a ''K''(''G'', 1).