Adjunction

Adjunction is ubiquitous in mathematics, as it specifies intuitive notions of optimization and efficiency.

Adjunctions in full There are hence numerous functors and natural transformations associated with every adjunction, and only a small portion is sufficient to determine the rest.

Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role of adjunction was inherent in Grothendieck's approach.

One can verify directly that this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).

The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification.

Then, a direct verification that they form a counit-unit adjunction is as follows: The first counit-unit equation says that for each set Y the composition : should be the identity.

Two constructions, called the category of Eilenberg–Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.