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 $ {\mathcal {C}} $ and $ {\mathcal {D}} $ is a pair of functors (assumed to be covariant) $ F : {\mathcal {D}} \rightarrow \mathcal {C} $ and $ G : {\mathcal {C}} \rightarrow {\mathcal {D}} $ and, for all objects $X$ in $ {\mathcal {C}} $ and $Y$ in $ {\mathcal {D}} $, a bijection between the respective morphism sets: $$ \mathrm {hom} _{\mathcal {C}} (FY, X) \cong \mathrm {hom} _{\mathcal {D}}(Y, GX) $$ such that this family of bijections is natural in $X$ and $Y$.

Naturality here means that there are natural isomorphisms between the pair of functors $ {\mathcal {C}}(F-,X) : {\mathcal {D}} \to \mathrm {Set} $ and $ {\mathcal {D}}(-,GX) : {\mathcal {D}} \to \mathrm {Set} $ for a fixed $X$ in $ {\mathcal {C}}$, and also the pair of functors $ {\mathcal {C}}(FY,-) : {\mathcal {C}} \to \mathrm {Set} $ and $ {\mathcal {D}}(Y,G-) : {\mathcal {C}} \to \mathrm {Set} $ for a fixed $Y$ in $ {\mathcal {D}} $.

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 $ {\mathcal {C}} $ and $ {\mathcal {D}} $ is somewhat akin to a "weak form" of an equivalence between $ {\mathcal {C}} $ and $ {\mathcal {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.

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Contents:
• 1 Terminology and notation
• 2 Introduction and Motivation
• 2.1 Solutions to optimization problems
• 2.2 Symmetry of optimization problems
• 3 Formal definitions
• 3.1 Conventions
• 3.2 Definition via universal morphisms
• 3.3 Definition via Hom-set adjunction
• 3.4 Definition via counit–unit adjunction
• 4 History
• 4.1 Ubiquity
• 5 Examples
• 5.1 Free groups
• 5.2 Free constructions and forgetful functors
• 5.3 Diagonal functors and limits
• 5.4 Colimits and diagonal functors
• 5.5 Further examples
• 5.5.1 Algebra
• 5.5.2 Topology
• 5.5.3 Posets
• 5.5.4 Category theory
• 5.5.5 Categorical logic
• 6 Adjunctions in full
• 6.1 Universal morphisms induce hom-set adjunction
• 6.2 counit–unit adjunction induces hom-set adjunction
• 6.3 Hom-set adjunction induces all of the above
• 7 Properties
• 7.1 Existence
• 7.2 Uniqueness
• 7.3 Composition
• 7.4 Limit preservation
• 7.5 Additivity
• 8 Relationships
• 8.1 Universal constructions
• 8.2 Equivalences of categories
• 8.3 Monads
• 9 Notes
• 10 References
• 11 External links

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