Editing Brown's representability theorem (section)

==Brown representability theorem for CW complexes==

The representability theorem for CW complexes, due to [[Edgar H. Brown]],<ref>{{Citation | last1=Brown | jstor=1970209| first1=Edgar H. | title=Cohomology theories | mr=0138104  | year=1962 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=75 | issue=3| pages=467–484 | doi=10.2307/1970209 }}</ref> is the following.  Suppose that:
# The functor ''F'' maps [[coproducts]] (i.e. [[wedge sum]]s) in ''Hotc'' to products in ''Set'': <math>F(\vee_\alpha X_\alpha) \cong \prod_\alpha F(X_\alpha),</math>
# The functor ''F'' maps [[Mapping cylinder|homotopy pushouts]] in ''Hotc'' to [[weak pullbacks]].  This is often stated as a [[Mayer–Vietoris sequence|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'') ≅ ''Hom''<sub>''Hotc''</sub>(''Z'', ''C'')
for any CW complex ''Z'', which is [[Natural transformation|natural]] in ''Z'' in that for any morphism from ''Z'' to another CW complex ''Y'' the induced maps ''F''(''Y'') → ''F''(''Z'') and ''Hom''<sub>''Hot''</sub>(''Y'', ''C'') → ''Hom''<sub>''Hot''</sub>(''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]].

Taking ''F''(''X'') to be the [[singular cohomology]] group ''H''<sup>''i''</sup>(''X'',''A'') with coefficients in a given abelian group ''A'', for fixed ''i'' > 0; then the representing space for ''F'' is the [[Eilenberg–MacLane space]] ''K''(''A'', ''i''). This gives a means of showing the existence of Eilenberg-MacLane spaces.