 Brown's representability theorem

A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by $[X,Y{]}_{\ast }$ for a CW complex $Y$. Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.

Brown's representability theorem is a result in category theory, specifically in the context of homological algebra and the study of functors. It provides criteria for when a covariant functor from a category of topological spaces (or more generally, from a category of 'nice' spaces) to the category of sets can be represented as the set of morphisms from a single object in a certain category. More precisely, the theorem addresses contravariant functors from topological spaces to sets.