Brown representability theorem

E886929

mathematical theorem result in category theory result in homotopy theory

The Brown representability theorem is a fundamental result in homotopy theory and category theory that characterizes when a contravariant functor from a homotopy category to sets (or abelian groups) is representable, providing a powerful tool for constructing and understanding cohomology theories.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in category theory ⓘ result in homotopy theory ⓘ
appliesTo	contravariant functors from homotopy categories to abelian groups ⓘ contravariant functors from homotopy categories to sets ⓘ
characterizes	when a contravariant functor is representable ⓘ
concerns	cohomology theories ⓘ contravariant functors ⓘ homotopy category ⓘ representable functors ⓘ
context	homotopy category of CW complexes ⓘ triangulated categories ⓘ
ensures	cohomology functors are representable ⓘ existence of representing objects for suitable functors ⓘ
field	algebraic topology ⓘ category theory ⓘ homotopy theory ⓘ
generalizationOf	classical representability results in category theory ⓘ
givesConditionOn	abelian-group-valued functors on homotopy categories ⓘ set-valued functors on homotopy categories ⓘ
hasVariant	Brown representability for homology ⓘ Brown representability in triangulated categories ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Eilenberg–Steenrod type representability results ⓘ existence of representing spaces for cohomology theories ⓘ
mathematicalArea	homological algebra ⓘ stable homotopy theory ⓘ
namedAfter	Edgar H. Brown Jr. NERFINISHED ⓘ
relatedTo	Eilenberg–Steenrod axioms NERFINISHED ⓘ Freyd adjoint functor theorem NERFINISHED ⓘ Yoneda lemma NERFINISHED ⓘ
requires	exactness with respect to homotopy cofiber sequences ⓘ wedge axiom for functors ⓘ
status	fundamental result in algebraic topology ⓘ standard tool in modern homotopy theory ⓘ
toolFor	constructing cohomology theories ⓘ understanding generalized cohomology theories ⓘ
type	representability theorem ⓘ
typicalCodomain	category of abelian groups ⓘ category of sets ⓘ
typicalDomain	homotopy category of pointed CW complexes ⓘ
usedIn	derived categories ⓘ spectra and generalized cohomology ⓘ stable homotopy theory ⓘ triangulated category theory ⓘ
usedToShow	generalized cohomology theories are represented by spectra ⓘ ordinary cohomology is represented by Eilenberg–Mac Lane spaces ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Atiyah–Hirzebruch spectral sequence → relatedConcept → Brown representability theorem ⓘ