Adjoint functors

Formal criteria for adjoint functors

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 โ€” 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 โ€” 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. (Wikipedia).

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From playlist The Distributive Property and Simplifying Algebraic Expressions

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From playlist Simplify Using the Rules of Exponents

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From playlist Properties of Exponents

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From playlist Simplify Using the Rules of Exponents

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From playlist Higher Algebra

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From playlist Simplify Using the Rules of Exponents

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From playlist Miscellaneous Questions

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From playlist Mathematics

Related pages

Limit (category theory) | Adjoint functors | Functor | Kan extension | Category theory