Freyd adjoint functor theorem
E621114
The Freyd adjoint functor theorem is a fundamental result in category theory that provides general conditions under which a functor admits a left or right adjoint, linking completeness and solution-set conditions to the existence of adjoint functors.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Freyd adjoint functor theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6834096 — 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: Freyd adjoint functor theorem Context triple: [Peter Freyd, notableWork, Freyd adjoint functor theorem]
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A.
Grothendieck category
A Grothendieck category is an abelian category with exact filtered colimits and a generator, providing a highly general framework that extends the properties of module and sheaf categories in homological algebra.
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B.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
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C.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
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D.
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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E.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
- 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: Freyd adjoint functor theorem Target entity description: The Freyd adjoint functor theorem is a fundamental result in category theory that provides general conditions under which a functor admits a left or right adjoint, linking completeness and solution-set conditions to the existence of adjoint functors.
-
A.
Grothendieck category
A Grothendieck category is an abelian category with exact filtered colimits and a generator, providing a highly general framework that extends the properties of module and sheaf categories in homological algebra.
-
B.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
C.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
D.
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.
-
E.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appearsIn | Peter Freyd’s work on abelian categories ⓘ |
| appliesTo |
functors preserving colimits
ⓘ
functors preserving limits ⓘ |
| concerns |
adjoint functors
ⓘ
existence of left adjoints ⓘ existence of right adjoints ⓘ functors between categories ⓘ |
| field | category theory ⓘ |
| givesConditionFor |
existence of left adjoint functor
ⓘ
existence of right adjoint functor ⓘ |
| hasVariant |
general adjoint functor theorem
NERFINISHED
ⓘ
special adjoint functor theorem ⓘ |
| implies | existence of adjoints under completeness and solution set conditions ⓘ |
| importance | fundamental existence theorem for adjoints in category theory ⓘ |
| isToolFor |
constructing adjoint functors
ⓘ
proving existence of free objects ⓘ proving existence of limits and colimits via adjoints ⓘ |
| namedAfter | Peter Freyd NERFINISHED ⓘ |
| relatedTo |
Brown representability theorem
NERFINISHED
ⓘ
Yoneda lemma NERFINISHED ⓘ adjunction between categories ⓘ |
| relatesConcept |
cocomplete categories
ⓘ
complete categories ⓘ completeness of categories ⓘ locally small categories ⓘ smallness conditions ⓘ solution set condition ⓘ |
| standardReference |
Peter Freyd – Abelian Categories
NERFINISHED
ⓘ
Saunders Mac Lane – Categories for the Working Mathematician NERFINISHED ⓘ |
| typicalConclusion |
functor has a left adjoint
ⓘ
functor has a right adjoint ⓘ |
| typicalHypothesis |
domain category is complete
ⓘ
functor preserves limits ⓘ solution set condition holds ⓘ |
| usedIn |
algebra
ⓘ
higher category theory ⓘ homological algebra ⓘ topology ⓘ |
| usesConcept |
comma categories
ⓘ
initial objects ⓘ representable functors ⓘ terminal objects ⓘ universal morphisms ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Freyd adjoint functor theorem Description of subject: The Freyd adjoint functor theorem is a fundamental result in category theory that provides general conditions under which a functor admits a left or right adjoint, linking completeness and solution-set conditions to the existence of adjoint functors.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.