right adjoint (changes) in nLab

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Idea

The right adjoint functor of a functor, if it exists, is one of two best approximations to a weak inverse of that functor. (The other best approximation is the functor's left adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a right adjoint, forming an adjoint equivalence.

A right adjoint to a forgetful functor is called a cofree functor; in general, right adjoints may be thought of as being defined cofreely, consisting of anything that works in an inverse, regardless of whether it’s needed.

The concept generalises immediately to enriched categories and in 2-categories.

Definitions

Given posets (or prosets) CC and DD and a monotone function U:C→DU: C \to D, a right adjoint of UU is a monotone function G:D→CG: D \to C such that

U(x)≤y⇔x≤G(y) U(x) \leq y \;\Leftrightarrow\; x \leq G(y)

for all xx in CC and yy in DD.

Given locally small categories CC and DD and a functor U:C→DU: C \to D, a right adjoint of UU is a functor G:D→CG: D \to C with a natural isomorphism between the hom-set functors

Hom D(U(−),−),Hom C(−,G(−)):C op×D→Set. Hom_D(U(-),-), Hom_C(-,G(-)): C^op \times D \to Set .

Given VV-enriched categories CC and DD and a VV-enriched functor U:C→DU: C \to D, a left adjoint of UU is a VV-enriched functor F:D→CF: D \to C with a VV-enriched natural isomorphism between the hom-object functors

Hom C(F(−),−),Hom D(−,U(−)):D op×C→Set. Hom_C(F(-),-), Hom_D(-,U(-)): D^op \times C \to Set .

Given categories CC and DD and a functor U:C→DU: C \to D, a right adjoint of UU is a functor G:D→CG: D \to C with natural transformations

ι:id C→U;G,ϵ:G;U→id D \iota: id_C \to U ; G,\; \epsilon: G ; U \to id_D

satisfying certain triangle identities.

Given a 2-category ℬ\mathcal{B}, objects CC and DD of ℬ\mathcal{B}, and a morphism U:C→DU: C \to D in ℬ\mathcal{B}, a right adjoint of UU is a morphism G:D→CG: D \to C with 22-morphisms

ι:id C→U;G,ϵ:G;U→id D \iota: id_C \to U ; G,\; \epsilon: G ; U \to id_D

satisfying the triangle identities.

Although it may not be immediately obvious, these definitions are all compatible.

Whenever GG is a right adjoint of UU, we have that UU is a left adjoint of GG.

Properties

Right adjoint functors preserve

• right adjoints preserve limits

• right adjoints preserve monomorphisms.

• limits;

• monomorphisms.

• See Galois connection for right adjoints of monotone functions.

• See adjoint functor for right adjoints of functors.

• See adjunction for right adjoints in 22-categories.

• See examples of adjoint functors for examples.

• parametric right adjoint