Common
Adjoint functors
Relationship between two functors abstracting many common constructions
Adjoint functors
In mathematics, specifically category theory, 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 𝒞 and 𝒟 is a pair of functors (assumed to be covariant) F:𝒟→𝒞G:𝒞→𝒟 and, for all objects c in 𝒞 and d in 𝒟, a bijection between the respective morphism sets hom𝒞(Fd,c)≅hom𝒟(d,Gc) such that this family of bijections is natural in c and d. For locally small categories...
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