Formal criteria for adjoint functors

rdf:langString Formal criteria for adjoint functors

rdf:langString Freyd's adjoint functor theorem

rdf:langString Kan criterion for the existence of a left adjoint

rdf:langString 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.

rdf:langString 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 .

rdf:langString 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.