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Concept
Brown's representability theorem
Summary
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the Hotc of pointed connected CW complexes, to the Set, to be a representable functor.
More specifically, we are given
F: Hotcop → Set,
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 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 of pointed sets; in other words the sets are also given a base point.
The representability theorem for CW complexes, due to Edgar H. Brown, is the following. Suppose that:
The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set:
The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer–Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F(U), v ∈ F(V) such that u and v restrict to the same element of F(U ∩ V), there is an element w ∈ F(W) restricting to u and v, respectively.
Then F is representable by some CW complex C, that is to say there is an isomorphism
F(Z) ≅ HomHotc(Z, C)
for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and HomHot(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 object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of Yoneda's lemma.
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