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 (3)
| Label | Occurrences |
|---|---|
| Freyd adjoint functor theorem canonical | 3 |
| adjoint functor theorems | 1 |
| general adjoint functor theorem | 1 |
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
ⓘ
linked to:
Freyd adjoint functor theorem
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 ⓘ |
| relatedTo |
Brown representability theorem
ⓘ
Yoneda lemma ⓘ adjunction between categories ⓘ |
| relatesConcept |
cocomplete categories
ⓘ
complete categories ⓘ completeness of categories ⓘ locally small categories ⓘ smallness conditions ⓘ solution set condition ⓘ |
| standardReference |
Peter Freyd – Abelian Categories
ⓘ
Saunders Mac Lane – Categories for the Working Mathematician ⓘ
linked to:
Categories for the Working Mathematician
|
| 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
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: 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 (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Freyd adjoint functor theorem
subject linked to:
Freyd
linked to: Freyd adjoint functor theorem