Adjoint

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous.

The equivalent symmetric definitions involving adjunctions and the symmetric language of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage of making this fact explicit.

We say that F is left adjoint to G, and G is right adjoint to F. Note that G may have itself a right adjoint that is quite different from F (see below for an example).

Conversely, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′.

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.

Let F and G be a pair of adjoint functors with unit η and co-unit ε (see the article on adjoint functors for the definitions).

This article provides several such definitions: * The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint.

This means that T is left adjoint to the forgetful functor U (see the section below on relation to adjoint functors).

Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Additivity If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G is also an additive functor and the hom-set bijections : are, in fact, isomorphisms of abelian groups.

Adjoint functors arise everywhere, in all areas of mathematics.

Adjoint property is discussed in more general context in Hofman & Morris (2007) (e.

Alternatively one can observe that the functor that for each group takes the underlying monoid (ignoring inverses) has a left adjoint.

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 pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant, an antitone Galois connection).

A particular case of this happens when a continuous functor admits a left adjoint.

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

Dually, if G is additive with a left adjoint F, then F is also additive.

Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint ) linear operator acting on the state space.

For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits.