In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, an Introduction to the Theory of Functors: Freyd's adjoint functor theorem — Let be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues): Another criterion is:
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| - Formal criteria for adjoint functors (en)
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| - In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, an Introduction to the Theory of Functors: Freyd's adjoint functor theorem — Let be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues): Another criterion is: (en)
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| - Freyd's adjoint functor theorem (en)
- Kan criterion for the existence of a left adjoint (en)
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| - In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor. One criterion is the following, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, an Introduction to the Theory of Functors: Freyd's adjoint functor theorem — Let be a functor between categories such that is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues): 1.
* G has a left adjoint. 2.
* preserves all limits and for each object x in , there exist a set I and an I-indexed family of morphisms such that each morphism is of the form for some morphism . Another criterion is: Kan criterion for the existence of a left adjoint — Let be a functor between categories. Then the following are equivalent. 1.
* G has a left adjoint. 2.
* G preserves limits and, for each object x in , the limit exists in . 3.
* The right Kan extension of the identity functor along G exists and is preserved by G. Moreover, when this is the case then a left adjoint of G can be computed using the left Kan extension. (en)
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| - Let be a functor between categories such that is complete. Then the following are equivalent :
# G has a left adjoint.
# preserves all limits and for each object x in , there exist a set I and an I-indexed family of morphisms such that each morphism is of the form for some morphism . (en)
- Let be a functor between categories. Then the following are equivalent.
# G has a left adjoint.
# G preserves limits and, for each object x in , the limit exists in .
# The right Kan extension of the identity functor along G exists and is preserved by G.
Moreover, when this is the case then a left adjoint of G can be computed using the left Kan extension. (en)
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