nLab
left adjoint

Contents

Idea

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

A left adjoint to a forgetful functor is called a free functor; in general, left adjoints may be thought of as being defined freely, consisting of anything that an inverse might want, regardless of whether it works.

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

Definitions

Given posets (or prosets) CC and DD and a monotone function U:CDU: C \to D, a left adjoint of UU is a monotone function F:DCF: D \to C such that

F(x)yxU(y) F(x) \leq y \;\Leftrightarrow\; x \leq U(y)

for all xx in CC and yy in DD.

Given locally small categories CC and DD and a functor U:CDU: C \to D, a left adjoint of UU is a functor F:DCF: D \to C with a natural isomorphism between the hom-set functors

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

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

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

Given categories CC and DD and a functor U:CDU: C \to D, a left adjoint of UU is a functor F:DCF: D \to C with natural transformations

ι:id CF;U,ϵ:U;Fid D \iota: id_C \to F ; U,\; \epsilon: U ; F \to id_D

(where F;UF;U etc gives the composite in the forwards, anti-Leibniz order) satisfying certain triangle identities.

Given a 2-category \mathcal{B}, objects CC and DD of \mathcal{B}, and a morphism U:CDU: C \to D in \mathcal{B}, a left adjoint of UU is a morphism F:DCF: D \to C with 22-morphisms

ι:id CF;U,ϵ:U;Fid D \iota: id_C \to F ; U,\; \epsilon: U ; F \to id_D

satisfying the triangle identities.

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

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

Properties

Left adjoint functors preserve

Further remarks

Revised on November 1, 2010 22:59:48 by Urs Schreiber (87.212.203.135)