In mathematics, Brown's representability theorem in homotopy theory[1] gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.
More specifically, we are given and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from category theory alone.
The converse statement also holds: any functor represented by a CW complex satisfies the above two properties.
This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.
A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman.