Adjoint functors

In mathematics, specifically , adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. 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 and is a pair of functors (assumed to be covariant) and and, for all objects in and in , a bijection between the respective morphism sets such that this family of bijections is natural in and . Naturality here means that there are natural isomorphisms between the pair of functors and for a fixed in , and also the pair of functors and for a fixed in . The functor is called a left adjoint functor or left adjoint to , while is called a right adjoint functor or right adjoint to . We write . An adjunction between categories and is somewhat akin to a "weak form" of an equivalence between and , 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. The terms adjoint and adjunct are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text Categories for the working mathematician, Mac Lane makes a distinction between the two. Given a family of hom-set bijections, we call an adjunction or an adjunction between and . If is an arrow in , is the right adjunct of (p. 81). The functor is left adjoint to , and is right adjoint to . (Note that may have itself a right adjoint that is quite different from ; see below for an example.