Adjunction (category theory)
In mathematics, specifically category theory, adjunction is a possible relationship between two functors.
Adjunction is ubiquitous in mathematics, as it specifies intuitive notions of optimization and efficiency.
In the most concise symmetric definition, an adjunction between categories C and D is a pair of functors,
and a family of bijections
which is natural for all variables X in C and Y in D. The functor F is called a left adjoint functor, while G is called a right adjoint functor. The relationship “F is left adjoint to G” (or equivalently, “G is right adjoint to F”) is sometimes written
This definition and others are made precise below.
The slogan is "Adjoint functors arise everywhere".
The long list of examples in this article is only a partial indication of how often an interesting mathematical construction is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence of their various definitions or the fact that they respectively preserve colimits/limits (which are also found in every area of mathematics), can encode the details of many useful and otherwise non-trivial results.
One can observe (e.g. in this article), two different roots are 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.
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