| Brown's Representability Theorem |
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Information AboutBrown's Representability Theorem |
F and certain necessary conditions for ''F'' to be of type ''Hom''(—, ''C'') with ''C'' a CW-complex can be deduced from Category Theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. The basic form of the conditions is that finite (coproduct of Pointed Space s) becomes under ''F'' a product of sets, and a Mayer-Vietoris axiom explaining the effect of ''F'' for patching, in terms of ''F''(''W'') where ''W'' is created by the identification of spaces ''U'' and ''V''. The statement of Brown's representability theorem is then that ''F'' is a representable functor on ''Hot'' (up to Equivalence Of Functors ) if and only if the wedge axiom and Mayer-Vietoris axiom are satisfied by ''F''. For an example, we can take ''F''(''X'') to be the Singular Cohomology group ''H''''i''(''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''). There is another version, applying the same ideas to the category of Spectra . |
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