On Brown representability theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:57:34Z http://mathoverflow.net/feeds/question/71812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71812/on-brown-representability-theorem On Brown representability theorem Yasha 2011-08-01T17:14:26Z 2011-08-01T20:48:45Z <p>The classical Brown representability theorem is for set valued functors. Is there a version for abelian group valued functors, and ring valued functors? </p> <p>In other words say we have an abelian group valued functor F on the category of CW top. spaces, satisfying the necessery condition that F maps colimits to limits. What extra conditions on F do we need to ensure that the classifying object is an H-space. Actually Brown doesn't state this, but at a brief glance his paper seems to prove that F just needs to satisfy excision that is we have exact sequences $$0 \to F (V \cap W) \to F(V) \oplus F (W) \to F (V \cup W) \to 0,$$ for V,W open sets in X. Is this right? What about the case of ring valued functors, when are they representable by (E_\infty? whatever that is)-ring space.</p> http://mathoverflow.net/questions/71812/on-brown-representability-theorem/71829#71829 Answer by Mark Grant for On Brown representability theorem Mark Grant 2011-08-01T20:48:45Z 2011-08-01T20:48:45Z <p>For the group case, this is an exercise in Chapter 9 of Switzer's "Algebraic Topology: Homotopy and Homology" (bottom of page 157).</p> <p>I think the rough idea is as follows. Suppose your functor $F$ on pointed CW complexes is representable as $F(-)=[-,Y\;]$, and takes values in the category of groups. You wish to show that $Y$ is a group-up-to-homotopy. To get a multiplication on $Y$ use the naturality part of Brown's theorem (Theorem 9.13 in Switzer). The functor $F\times F$ is represented by $Y\times Y$, and the group multiplication is a natural transformation $F\times F\to F$, which is therefore represented by a map $m\colon\thinspace Y\times Y\to Y$, unique up to homotopy.</p> <p>Now check that the group axioms for $F(-)$ yield the desired properties of $m$ (associative, unital, with inverses up to homotopy). </p>