Brown's Representability Theorem - Variants

Variants

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak 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 to connected pointed spaces, and an analogous statement for unpointed spaces is also false.

A similar statement does, however, hold for spectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem, which includes both the version for pointed connected CW complexes and the version for spectra.

A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman. Together with the preceding remark, it gives a criterion for a (covariant) functor F: C → D between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a left adjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

Jacob Lurie has proved a version of the Brown representability theorem for the homotopy category of a pointed quasicategory 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 unbounded derived category of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).