nLab
adjoint functor

Definition

We say that two functors $L:C\to D$ and $R:D\to C$ are adjoint if they form an adjunction $L⊣R$ in the 2-category Cat of categories. This means that they are equipped with natural transformations $\eta :1_C\to R\circ L$ and $ϵ:L\circ R\to 1_D$ satisfying the triangle identities, that is the compositions $L\stackrel{L\eta }{\to }LRL\stackrel{ϵL}{\to }L$ and $R\stackrel{\eta R}{\to }RLR\stackrel{Rϵ}{\to }R$ are identities. The left or right adjoint of any functor, if it exists, is unique up to unique isomorphism.

We say that $L$ is the left adjoint of $R$ and that $R$ is the right adjoint of $L$.

In the case of Cat, there are a number of equivalent characterizations of an adjunction, some of which are given below.

Definition in terms of Hom isomorphism

An adjunction $L⊣R$ is equivalently given by a natural isomorphism of hom-functors $C^{\mathrm{op}}\times D\to \mathrm{Set}$

$${\mathrm{Hom}}_D(L(-),-)\simeq {\mathrm{Hom}}_C(-,R(-)).$$Hom_D(L(-),-) \simeq Hom_C(-,R(-)) \,.

In other words, for all $c\in C$ and $d\in D$ there is a bijection of sets

$${\mathrm{Hom}}_D(L(c),d)\simeq {\mathrm{Hom}}_C(c,R(d))$$Hom_D(L(c),d) \simeq Hom_C(c,R(d))

naturally in $c$ and $d$. This isomorphism is the adjunction isomorphism and the image of an element under this isomorphism is its adjunct.

Given such an adjunction isomorphism, $\eta$ and $ϵ$ can be recovered as the adjuncts of identity morphisms. The Yoneda lemma ensures that the entire adjunction isomorphism can be recovered from them by composition: the adjunct of $f:L(c)\to d$ is $R(f)\eta$, and the adjunct of $g:c\to R(d)$ is $ϵL(g)$. The triangle identities are precisely what is necessary to ensure that this is an isomorphism.

Definition in terms of representable functors

A functor $L:C\to D$ has a right adjoint if and only if for all $d$, the presheaf ${\mathrm{Hom}}_D(L(-),d):D^{\mathrm{op}}\to \mathrm{Set}$ is representable, i.e. there exists an object $R(d)$ and a natural isomorphism

$${\mathrm{Hom}}_D(L(-),d)\cong {\mathrm{Hom}}_C(-,R(d)).$$Hom_D(L(-),d) \cong Hom_C(-,R(d)).

There is then a unique way to define $R$ on arrows so as to make these isomorphisms natural in $d$ as well.

In more fancy language, by precomposition $L$ defines a functor

$$L^{*}:[D^{\mathrm{op}},\mathrm{Set}]\to [C^{\mathrm{op}},\mathrm{Set}]$$L^* : [D^{op}, Set] \to [C^{op}, Set]

of presheaf categories. By restriction along the Yoneda embedding $Y:D\to [D^{\mathrm{op}},\mathrm{Set}]$ this yields the functor

$$\overline{L}:D\stackrel{Y}{\to }[D^{\mathrm{op}},\mathrm{Set}]\stackrel{L^{*}}{\to }[C^{\mathrm{op}},\mathrm{Set}].$$\bar L : D \stackrel{Y}{\to} [D^{op}, Set] \stackrel{L^*}{\to} [C^{op}, Set] \,.

such that

$$\overline{L}:d\mapsto {\mathrm{Hom}}_D(L(-),d)\in [D^{\mathrm{op}},\mathrm{Set}].$$\bar L : d \mapsto Hom_D(L(-),d) \in [D^{op}, Set] \,.

If for all $d\in D$ this presheaf $\overline{L}(d)$ is representable, then it is functorially so in that there exists a functor $R:D\to C$ such that

$$\overline{L}\simeq Y\circ R.$$\bar L \simeq Y \circ R \,.

Local definition

This definition has the advantage that it yields useful information even if the adjoint functor $R$ does not exist globally, i.e. as a functor on all of $D$:

it may happen that

$$\overline{L}(d):={\mathrm{Hom}}_D(L(-),d)\in [D^{\mathrm{op}},\mathrm{Set}]$$\bar L(d) := Hom_D(L(-),d) \in [D^{op}, Set]

is representable for some $d$ but not for all $d$. The representing object may still usefully be thought of as $R(d)$.

This global versus local evaluation of adjoint functors induces the global/local pictures of the defintions

• limit / homotopy limit

• Kan extension

as discussed there.

Definition in terms of universal arrows

Given $R:D\to C$, and $c\in C$, a universal arrow from $c$ to $R$ is an initial object of the comma category $(c/R)$. That is, it consists of an object $L(c)\in D$ and an arrow $\eta :c\to R(L(c))$ such that for any $d\in D$, any arrow $g:c\to R(d)$ factors as $R(f)\eta$ for a unique $f:L(c)\to d$. In particular, we have a bijection

$${\mathrm{Hom}}_C(c,R(d))\cong {\mathrm{Hom}}_D(L(c),d)$$Hom_C(c,R(d)) \cong Hom_D(L(c),d)

which it is easy to see is natural in $d$. Again, in this case there is a unique way to make $L$ into a functor so that this isomorphism is natural in $c$ as well.

Note that this definition is simply obtained by applying the Yoneda lemma to the definition in terms of representable functors.

Definition in terms of correspondences

Every distributor

$$k:C^{\mathrm{op}}\times D\to S$$k : C^{op} \times D \to S

defines a category $C{*}^{k}D$ with $\mathrm{Obj}(C{*}^{k}D)=\mathrm{Obj}(C)\bigsqcup \mathrm{Obj}(D)$ and

$${\mathrm{Hom}}_{C^{\mathrm{op}}\times D}(X,Y)=\{\begin{array}{cc}{\mathrm{Hom}}_C(X,Y)& \mathrm{if}X,Y\in C\\ {\mathrm{Hom}}_D(X,Y)& \mathrm{if}X,Y\in D\\ k(X,Y)& \mathrm{if}X\in C\mathrm{and}Y\in D\\ \varnothing & \mathrm{otherwise}\end{array}.$$Hom_{C^{op} \times D}(X,Y) = \left\{ \array{ Hom_C(X,Y) & if X, Y \in C \\ Hom_{D}(X,Y) & if X,Y \in D \\ k(X,Y) & if X \in C and Y \in D \\ \emptyset & otherwise } \right. \,.

This category naturally comes with a functor to the interval category

$$C{*}^{k}D\to {\Delta }^1.$$C *^k D \to \Delta^1 \,.

Now, every functor $L:C\to D$ induces a distributor

$$k_f(X,Y)={\mathrm{Hom}}_D(f(X),Y)$$k_f(X,Y) = Hom_D(f(X), Y)

and every functor $R:D\to C$ induces a distributor

$$k_g(X,Y)={\mathrm{Hom}}_C(X,R(Y)).$$k_g(X,Y) = Hom_C(X, R(Y)) \,.

The functors $L$ and $R$ are adjoint precisely if the distributors that they define in the above way are equal. This in turn is the case if $C{\star }^{L}D\simeq (D^{\mathrm{op}}{\star }^{R^{\mathrm{op}}}{C}^{\mathrm{op}}{)}^{\mathrm{op}}$.

We say that $C{\star }^{k}D$ is the cograph of the functor $k$. See there for more on this.

Definition for $(\mathrm{\infty }\mathrm{,1})$-functors

The above characterization of adjoint functors in terms of categories over the interval is used in section 5.2.2 of

• Jacob Lurie, Higher Topos Theory

(motivated from the discussion of correspondences in section 2.3.1)

to give a definition of adjunction between (infinity,1)-functors.

Definition

Let $C$ and $D$ be quasi-categories. An adjunction between $C$ and $D$ is

• a morphism of simplicial sets $K\to {\Delta }^1$

• which is

  • a Cartesian fibration

  • a coCartesian fibration

• such that $C\simeq K_0$ and $D\simeq K_1$.

For more on this see

• adjoint (∞,1)-functor.

Examples

• see examples of adjoint functors.

Properties

Let $L⊣R$ be a pair of adjoint functors. We have the following

• ($R$ is full and faithful) $\iff$ ($ϵ:L\circ R\stackrel{\simeq }{\to }{\mathrm{Id}}_D$)

• ($L$ is full and faithful) $\iff$ ($\eta :{\mathrm{Id}}_C\stackrel{\simeq }{\to }R\circ L$)

• the following are equivalent

  • $L$ and $R$ are both full and faithful;

  • $L$ is an equivalence;

  • $R$ is an equivalence.

• $L$ preserves all colimits that may exist in $C$, while $R$ preserves all limits in $D$. For a partial converse, see the adjoint functor theorem.

Videos

• The Catsters list