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 | 4 |
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.
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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. ⓘ |
| relatedTo |
Eilenberg–Steenrod axioms
ⓘ
Freyd adjoint functor theorem ⓘ Yoneda lemma ⓘ |
| 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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (4)
Full triples — surface form annotated when it differs from this entity's canonical label.