Editing Brown's representability theorem (section)

{{short description|On representability of a contravariant functor on the category of connected CW complexes}}
In mathematics, '''Brown's representability theorem''' in [[homotopy theory]]<ref>{{Citation | last1=Switzer | first1=Robert M. | title=Algebraic topology---homotopy and homology | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 | mr=1886843  | year=2002 | pages = 152–157}}</ref> gives [[necessary and sufficient condition]]s for a [[contravariant functor]] ''F'' on the [[homotopy category]] ''Hotc'' of pointed connected [[CW complex]]es, to the [[category of sets]] '''Set''', to be a [[representable functor]].

More specifically, we are given

:''F'': ''Hotc''<sup>op</sup> → '''Set''',

and there are certain obviously necessary conditions for ''F'' to be of type ''Hom''(&mdash;, ''C''), with ''C'' a pointed connected CW-complex that can be deduced from [[category theory]] alone.  The statement of the substantive part of the theorem is that these necessary conditions are then sufficient.  For technical reasons, the theorem is often stated for functors to the category of [[pointed set]]s; in other words the sets are also given a base point.