In mathematics,Brown's representability theorem inhomotopy theory[1] givesnecessary and sufficient conditions for acontravariant functorF on thehomotopy categoryHotc of pointed connectedCW complexes, to thecategory of setsSet, to be arepresentable functor.
More specifically, we are given
and there are certain obviously necessary conditions forF to be of typeHom(—,C), withC a pointed connected CW-complex that can be deduced fromcategory 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 ofpointed sets; in other words the sets are also given a base point.
The representability theorem for CW complexes, due toEdgar H. Brown,[2] is the following. Suppose that:
ThenF is representable by some CW complexC, that is to say there is an isomorphism
for any CW complexZ, which isnatural inZ in that for any morphism fromZ to another CW complexY the induced mapsF(Y) →F(Z) andHomHot(Y,C) →HomHot(Z,C) are compatible with these isomorphisms.
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.
The representing objectC above can be shown to depend functorially onF: anynatural transformation fromF to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence ofYoneda's lemma.
TakingF(X) to be thesingular cohomology groupHi(X,A) with coefficients in a given abelian groupA, for fixedi > 0; then the representing space forF is theEilenberg–MacLane spaceK(A,i). This gives a means of showing the existence of Eilenberg-MacLane spaces.
Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at theweak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.
However, the theorem is false without the restriction toconnected pointed spaces, and an analogous statement for unpointed spaces is also false.[3]
A similar statement does, however, hold forspectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem,[4] which includes both the version for pointed connected CW complexes and the version for spectra.
A version of the representability theorem in the case oftriangulated categories is due to Amnon Neeman.[5] Together with the preceding remark, it gives a criterion for a (covariant) functorF:C →D between triangulated categories satisfying certain technical conditions to have a rightadjoint functor. Namely, ifC andD are triangulated categories withC compactly generated andF a triangulated functor commuting with arbitrary direct sums, thenF is a left adjoint. Neeman has applied this to proving theGrothendieck duality theorem in algebraic geometry.
Jacob Lurie has proved a version of the Brown representability theorem[6] for the homotopy category of a pointedquasicategory with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of pointed connected CW complexes, as well as to the unboundedderived category of aGrothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).