Theorem

Brown Representability Theorem

Topology

The Brown Representability Theorem gives conditions under which a contravariant functor from the homotopy category of pointed, connected CW complexes to the category of sets is representable, meaning it agrees with the functor of homotopy classes of maps into some fixed space, provided the given functor sends wedge sums to products and satisfies a gluing condition analogous to the Mayer-Vietoris sequence. Named for Edgar Brown, it is a foundational existence result of algebraic topology guaranteeing that many naturally arising cohomology-like theories are represented by an actual topological space, called a classifying space.

Facts

Classification

Statement Form

Characterization Theorem 1

Proof Year

1962 1

Connections

Has Statement Form

Characterization

Proved By

Edgar H. Brown

Source Brown's representability theorem (Wikipedia)

Sources

1. Brown representability theorem (Wikipedia)

Introduction

Quote, Introduction

The theorem is due to Edgar H. Brown who published it in 1962.

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Brown's representability theorem (Wikipedia)

Proved By: Edgar H. Brown, Lead
The theorem is due to Edgar H. Brown who published it in 1962.
In Group: Algebraic Topology, Lead paragraph
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

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Brown Representability Theorem

Has Statement Form

Proved By