Freyd adjoint functor theorem

E621114

mathematical theorem

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.

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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 ⓘ

Referenced by (2)

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

Peter Freyd → notableWork → Freyd adjoint functor theorem ⓘ

Peter Freyd → knownFor → Freyd adjoint functor theorem ⓘ