structures in a cohesive (∞,1)-topos
The classical Brown representability theorem says that certain contravariant functors on the homotopy category are representable. This is used, for instance, to show that any generalized cohomology theory is representable by a spectrum.
the existence of a strong generator in the homotopy category in question.
In particular, there is no Brown representability theorem for functors from the homotopy category of pointed not-necessarily-connected spaces to pointed sets, or for functors from the homotopy category of unpointed spaces to sets. In fact, there are counterexamples in these two cases.
Let denote the homotopy category of pointed, connected, topological spaces under weak homotopy equivalence — or equivalently the homotopy category of pointed connected CW complexes under homotopy equivalence. Then has coproducts (given by wedge sums) and also weak pushouts (namely, homotopy pushouts).
(Brown) If a functor takes coproducts to products, and weak pushouts to weak pullbacks, then it is representable. That is, there is a pointed connected CW-complex and a universal element such that is a natural isomorphism.
Note that it is immediate that every representable functor has the given properties; the nontrivial statement is that these properties already characterize representable functors.
When the theorem is stated in terms of CW complexes, the second property (taking weak pushouts to weak pullbacks) is often phrased equivalently as:
For example, could be the homotopy category of pointed spaces (under weak homotopy equivalence), with the full subcategory of (spaces of the homotopy type of) finite CW complexes. Motivated by this example, we write for hom-sets in .
Let denote the full subcategory of those such that for any , if the induced map is bijective for all , then is an isomorphism. If as above, then is the category of pointed connected spaces.
(Brown) With as above, let be a functor which takes coproducts to products and weak pushouts to weak pullbacks. Then there exists and a natural transformation such that
(Jardine) Let be a homotopical functor to the category of pointed sets on such that
preserves small coproducts of cofibrant objects (including preserving the zero object).
(Mayer-Vietoris property) takes any pushout diagram
with all objects cofibrant and a cofibration, to a weak pullback.
Then is representable.
This follows essentially immediately from Theorem 2 applied to . Jardine also proves a more refined version (see references).
In some places, one can find the classical Brown representability stated without the restricted to connected pointed spaces. However, this version is false, as is the analogous statement for unpointed spaces.
In the paper of Freyd-Heller cited below, it is show that if is Thompson's group F, with its canonical endomorphism, then does not split in the quotient of Grp by conjugacy. Since the quotient of Grp by conjugacy embeds as the full subcategory of the unpointed homotopy category on connected homotopy 1-types, we have an endomorphism of the classifying space of which does not split in .
Thus, if splits the idempotent of , then satisfies the hypotheses of the Brown representability theorem (being a retract of a representable functor), but is not representable. A similar argument using applies to .
There is also another example due to Heller, which fais to be representable for cardinality reasons.
The original theorem was proven in
The categorical generalization was proven in
The model-categorical version, with applications to ∞-stacks – or rather to the standard models for ∞-stack (∞,1)-toposes in terms of the standard model structure on simplicial presheaves – is given in
The counterexamples to nonconnected and unpointed Brown representability are from
A review in the context of chromatic homotopy theory is in