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Adjoint functors

A pair of functors F ⊣ G equipped with a natural hom-set bijection (equivalently, a unit and counit satisfying the triangle identities).

Adjoint functors

Let $\mathcal C,\mathcal D$ be a category and let $F:\mathcal C\to\mathcal D$ and $G:\mathcal D\to\mathcal C$ be functors .

Definition (Adjunction)

We say $F$ is left adjoint to $G$ , and write $F \dashv G$ , if for every object $c\in\mathcal C$ and $d\in\mathcal D$ there is a bijection of sets

\Phi_{c,d}:\operatorname{Hom}_{\mathcal D}(F c,\, d)\;\xrightarrow{\;\cong\;}\;\operatorname{Hom}_{\mathcal C}(c,\, G d),

which is natural in $c$ and $d$ (i.e. it is a natural isomorphism of bifunctors $\operatorname{Hom}_{\mathcal D}(F-, -)\cong \operatorname{Hom}_{\mathcal C}(-, G-)$ ).

Elements of $\operatorname{Hom}$ are morphisms ; the naturality condition says that $\Phi$ commutes with pre- and post-composition in the two variables (using composition in $\mathcal C,\mathcal D$ ).

Equivalent data: unit and counit

An adjunction $F\dashv G$ is equivalently specified by:

a unit $\eta:\mathrm{Id}_{\mathcal C}\Rightarrow G F$ , a natural transformation (see unit ),
a counit $\varepsilon:F G\Rightarrow \mathrm{Id}_{\mathcal D}$ , a natural transformation (see counit ),

such that the triangle identities hold for all $c\in\mathcal C$ , $d\in\mathcal D$ :

\varepsilon_{F c}\circ F(\eta_c)=\mathrm{id}_{F c}, \qquad G(\varepsilon_d)\circ \eta_{G d}=\mathrm{id}_{G d}.

(Here $\mathrm{id}$ is the identity morphism .)

If moreover $\eta$ and $\varepsilon$ are natural isomorphisms , then $F$ and $G$ form an equivalence of categories .

Examples

Free/forgetful (Set–Grp). The free group functor $F:\mathbf{Set}\to\mathbf{Grp}$ is left adjoint to the forgetful functor $U:\mathbf{Grp}\to\mathbf{Set}$ . Concretely,
$\operatorname{Hom}_{\mathbf{Grp}}(F(X),\,H)\cong \operatorname{Hom}_{\mathbf{Set}}(X,\,U(H)),$
naturally in a set $X$ and a group $H$ .
Product–exponential (Set). Fix a set $X$ . In $\mathbf{Set}$ , the functor $(-)\times X$ is left adjoint to the exponential functor $(-)^X=\operatorname{Hom}(X,-)$ :
$\operatorname{Hom}(A\times X,\,B)\cong \operatorname{Hom}(A,\,B^X),$
naturally in $A,B\in\mathbf{Set}$ . (This is the usual currying bijection.)
Abelianization–inclusion (Grp–Ab). Let $i:\mathbf{Ab}\hookrightarrow \mathbf{Grp}$ be the inclusion. The abelianization functor $\mathrm{ab}:\mathbf{Grp}\to\mathbf{Ab}$ , $G\mapsto G/[G,G]$ , is left adjoint to $i$ :
$\operatorname{Hom}_{\mathbf{Ab}}(\mathrm{ab}(G),\,A)\cong \operatorname{Hom}_{\mathbf{Grp}}(G,\,i(A)).$