Brown representability theorem
E886929
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Brown representability theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829397 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Brown representability theorem Context triple: [Atiyah–Hirzebruch spectral sequence, relatedConcept, Brown representability theorem]
-
A.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
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B.
Freyd–Mitchell embedding theorem
The Freyd–Mitchell embedding theorem is a fundamental result in category theory stating that every small abelian category can be faithfully represented as a full subcategory of a module category, thereby allowing the use of element-wise methods in abstract settings.
-
C.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
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D.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
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E.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Brown representability theorem Target entity description: 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.
-
A.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
-
B.
Freyd–Mitchell embedding theorem
The Freyd–Mitchell embedding theorem is a fundamental result in category theory stating that every small abelian category can be faithfully represented as a full subcategory of a module category, thereby allowing the use of element-wise methods in abstract settings.
-
C.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
-
D.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
-
E.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Brown representability theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.