Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of 'optimal solutions' to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone-Čech compactification of a topological space in topology.The slogan is 'Adjoint functors arise everywhere'. In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of 'optimal solutions' to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone-Čech compactification of a topological space in topology. By definition, an adjunction between categories C and D is a pair of functors (assumed to be covariant) and, for all objects X in C and Y in D a bijection between the respective morphism sets such that this family of bijections is natural in X and Y. The functor F is called a left adjoint functor or left adjoint to G , while G is called a right adjoint functor or right adjoint to F . An adjunction between categories C and D is somewhat akin to a 'weak form' of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations, an adjunction can be 'upgraded' to an equivalence, by a suitable natural modification of the involved categories and functors. Two different roots are being used: 'adjunct' and 'adjoint'. From Oxford shorter English dictionary, 'adjunct' is from Latin, 'adjoint' is from French. In Mac Lane, Categories for the working mathematician, chap. 4, 'Adjoints', one can verify the following usage. Given a family φ X Y : h o m C ( F Y , X ) ≅ h o m D ( Y , G X ) {displaystyle varphi _{XY}:mathrm {hom} _{mathcal {C}}(FY,X)cong mathrm {hom} _{mathcal {D}}(Y,GX)} of hom-set bijections, we call φ {displaystyle varphi } an 'adjunction' or an 'adjunction between F {displaystyle F} and G {displaystyle G} '. If f {displaystyle f} is an arrow in h o m C ( F Y , X ) {displaystyle mathrm {hom} _{mathcal {C}}(FY,X)} , φ f {displaystyle varphi f} is the right 'adjunct' of f {displaystyle f} (p. 81). The functor F {displaystyle F} is left 'adjoint' to G {displaystyle G} and G {displaystyle G} is right adjoint to F {displaystyle F} . (Note that G may have itself a right adjoint that is quite different from F; see below for an example.)