category theory

Contents

Definition

The concept of adjoint functors is a key concept in category theory, if not the key concept.1 It embodies the concept of representable functors and has as special cases universal constructions such as Kan extensions and hence of limits/colimits.

More abstractly, the concept of adjoint functors is itself just the special case of the general concept of an adjunction in a 2-category, here for the 2-category Cat of all categories. But often “adjunction” is understood by default in this special case.

There are various different but equivalent characterizations of adjoint functors, some of which are discussed below.

In terms of Hom isomorphism

We discuss here the definition of adjointness of functors $L \dashv R$ in terms of a natural bijection between hom-sets (Def. below):

$\{L(c) \to d\} \;\simeq\; \{ c \to R(d) \}$

We show that this is equivalent to the abstract definition, in terms of an adjunction in the 2-category Cat, in Prop. below.

$\,$

Definition

(adjoint functors in terms of natural bijections of hom-sets)

Let $\mathcal{C}$ and $\mathcal{D}$ be two categories, and let

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}$

be a pair of functors between them, as shown. Then this is called a pair of adjoint functors (or an adjoint pair of functors) with $L$ left adjoint and $R$ right adjoint, denoted

if there exists a natural isomorphism between the hom-functors of the following form:

(1)$Hom_{\mathcal{D}}(L(-),-) \;\simeq\; Hom_{\mathcal{C}}(-,R(-)) \,.$

This means that for all objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there is a bijection of hom-sets

$\array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ ( L(c) \overset{f}{\to} d ) &\mapsto& (c \overset{\widetilde f}{\to} R(d)) }$

which is natural in $c$ and $d$. This isomorphism is the adjunction isomorphism and the image $\widetilde f$ of a morphism $f$ under this bijections is called the adjunct of $f$. Conversely, $f$ is called the adjunct of $\widetilde f$.

Naturality here means that for every morphism $g \colon c_2 \to c_1$ in $\mathcal{C}$ and for every morphism $h\colon d_1\to d_2$ in $\mathcal{D}$, the resulting square

(2)$\array{ Hom_{\mathcal{D}}(L(c_1), d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1, R(d_1)) \\ {}^{\mathllap{Hom_{\mathcal{D}}(L(g), h)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(g, R(h))}} \\ Hom_{\mathcal{D}}(L(c_2),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d_2)) }$

commutes (see also at hom-functor for the definition of the vertical maps here).

Explicitly, this commutativity, in turn, means that for every morphism $f \;\colon\; L(c_1) \to d_1$ with adjunct $\widetilde f \;\colon\; c_1 \to R(d_1)$, the adjunct of the composition is

$\array{ L(c_1) & \overset{f}{\longrightarrow} & d_1 \\ {}^{\mathllap{L(g)}}\big\uparrow && \big\downarrow^{\mathrlap{h}} \\ L(c_2) && d_2 } \;\;\;=\;\;\; \array{ c_1 &\overset{\widetilde f}{\longrightarrow}& R(d_1) \\ {}^{\mathllap{g}}\big\uparrow && \big\downarrow^{\mathrlap{R(h)}} \\ c_2 && R(d_2) }$
Definition

(adjunction unit and counit in terms of hom-isomorphism)

Given a pair of adjoint functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}$

according to Def. one says that

1. for any $c \in \mathcal{C}$ the adjunct of the identity morphism on $L(c)$ is the unit morphism of the adjunction at that object, denoted

$\eta_c \coloneqq \widetilde{id_{L(c)}} \;\colon\; c \longrightarrow R(L(c))$
2. for any $d \in \mathcal{D}$ the adjunct of the identity morphism on $R(d)$ is the counit morphism of the adjunction at that object, denoted

$\epsilon_d \;\colon\; L(R(d)) \longrightarrow d$
Proposition

(general adjuncts in terms of unit/counit)

Consider a pair of adjoint functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}$

according to Def. , with adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ according to Def. .

Then

1. The adjunct $\widetilde f$ of any morphism $L(c) \overset{f}{\to} d$ is obtained from $R$ and $\eta_c$ as the composite

(3)$\widetilde f \;\colon\; c \overset{\eta_c}{\longrightarrow} R(L(c)) \overset{R(f)}{\longrightarrow} R(d)$

Conversely, the adjunct $f$ of any morphism $c \overset{\widetilde f}{\longrightarrow} R(d)$ is obtained from $L$ and $\epsilon_d$ as

(4)$f \;\colon\; L(c) \overset{L(\widetilde f)}{\longrightarrow} L(R(d)) \overset{\epsilon_d}{\longrightarrow} d$
2. The adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ are components of natural transformations of the form

$\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L$

and

$\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}$
3. The adjunction unit and adjunction counit satisfy the triangle identities, saying that

$id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)$

and

$id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)$
Proof

For the first statement, consider the naturality square (2) in the form

$\array{ id_{L(c)} \in & Hom_{\mathcal{D}}(L(c), L(c)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c, R(L(c))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(id), f)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(id, R(f))}} \\ & Hom_{\mathcal{D}}(L(c), d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}( c, R(d) ) }$

and consider the element $id_{L(c_1)}$ in the top left entry. Its image under going down and then right in the diagram is $\widetilde f$, by Def. . On the other hand, its image under going right and then down is $R(f)\circ \eta_{c}$, by Def. . Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown, for the adjunct of $f$.

The converse formula follows analogously.

The third statement follows directly from this by applying these formulas for the adjuncts twice and using that the result must be the original morphism:

\begin{aligned} id_{L(c)} & = \widetilde \widetilde { id_{L(c)} } \\ & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) } \\ & = L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c) \end{aligned}

For the second statement, we have to show that for every morphism $f \colon c_1 \to c_2$ the following square commutes:

$\array{ c_1 &\overset{f}{\longrightarrow}& c_2 \\ {}^{\mathllap{\eta_{c_1}}}\big\downarrow && \big\downarrow^{\mathrlap{\eta_{c_2}}} \\ R(L(c_1)) &\underset{ R(L(f)) }{\longrightarrow}& R(L(c_2)) }$

To see this, consider the naturality square (2) in the form

$\array{ id_{L(c_2)} \in & Hom_{\mathcal{D}}(L(c_2), L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2, R(L(c_2))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(f),id_{L(c_2)})}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(f, R(id_{L(c_2)}))}} \\ & Hom_{\mathcal{D}}(L(c_1),L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(L(c_2))) }$

The image of the element $id_{L(c_2)}$ in the top left along the right and down is $\eta_{c_2} \circ f$, by Def. , while its image down and then to the right is $\widetilde{L(f)} = R(L(f)) \circ \eta_{c_1}$, by the previous statement. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown.

The argument for the naturality of $\epsilon$ is directly analogous.

Proposition

(adjointness in terms of hom-isomorphism equivalent to adjunction in $Cat$)

Two functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}$

are an adjoint pair in the sense that there is a natural isomorphism (1) according to Def. , precisely if they participate in an adjunction in the 2-category Cat, meaning that

1. there exist natural transformations

$\eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L$

and

$\epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}$
2. which satisfy the triangle identities

$id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)$

and

$id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)$
Proof

That a hom-isomorphism (1) implies units/counits satisfying the triangle identities is the statement of the second two items of Prop. .

Hence it remains to show the converse. But the argument is along the same lines as the proof of Prop. : We now define forming of adjuncts by the formula (3). That the resulting assignment $f \mapsto \widetilde f$ is an isomorphism follows from the computation

\begin{aligned} \widetilde {\widetilde f} & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) \overset{R(f)}{\to} R(d) } \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{L(R(f))}{\to} L(R(d)) \overset{\epsilon_d}{\to} d \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{ \epsilon_{L(c)} }{\to} L(c) \overset{f}{\longrightarrow} d \\ & = L(c) \overset{f}{\longrightarrow} d \end{aligned}

where, after expanding out the definition, we used naturality of $\epsilon$ and then the triangle identity.

Finally, that this construction satisfies the naturality condition (2) follows from the functoriality of the functors involved, and the naturality of the unit/counit:

$\array{ c_2 &\overset{ \eta_{c_2} }{\longrightarrow}& R(L(c_2)) \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{R(L(g))}} & \searrow^{\mathrlap{ R( L(g) \circ f ) }} \\ c_1 &\overset{\eta_{c_1}}{\longrightarrow}& R(L(c_1)) &\overset{R(f)}{\longrightarrow}& R(d_1) \\ && & {}_{R( h\circ f)}\searrow & \downarrow^{\mathrlap{ R(h) }} \\ && && R(d_2) }$

In terms of representable functors

The condition (1) on adjoint functors $L \dashv R$ in Def. implies in particular that for every object $d \in \mathcal{D}$ the functor $Hom_{\mathcal{D}}(L(-),d)$ is a representable functor with representing object $R(d)$. The following Prop. observes that the existence of such representing objects for all $d$ is, in fact, already sufficient to imply that there is a right adjoint functor.

This equivalent perspective on adjoint functors makes manifest that:

1. adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. below;

2. the concept of adjoint functors makes sense also relative to a full subcategory on which representing objects exists, this is the content of Remark below.

Global definition

Proposition

(adjoint functor from objectwise representing object)

A functor $L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a right adjoint $R \;\colon\; \mathcal{D} \to \mathcal{C}$, according to Def. , already if for all objects $d \in \mathcal{D}$ there is an object $R(d) \in \mathcal{C}$ such that there is a natural isomorphism

$Hom_{\mathcal{D}}(L(-),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(-,R(d)) \,,$

hence for each object $c \in \mathcal{C}$ a bijection

$Hom_{\mathcal{D}}(L(c),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(c,R(d))$

such that for each morphism $g \;\colon\; c_2 \to c_1$, the following diagram commutes

(5)$\array{ Hom_{\mathcal{D}}(L(c_1),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(d)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}(L(g),id_d) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, id_{R(d)} ) }} \\ Hom_{\mathcal{D}}(L(c_2),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d)) }$

(This is as in (2), except that only naturality in the first variable is required.)

In this case there is a unique way to extend $R$ from a function on objects to a function on morphisms such as to make it a functor $R \colon \mathcal{D} \to \mathcal{C}$ which is right adjoint to $L$. , and hence the statement is that with this, naturality in the second variable is already implied.

Proof

Notice that

1. in the language of presheaves the assumption is that for each $d \in \mathcal{D}$ the presheaf

$Hom_{\mathcal{D}}(L(-),d) \;\in\; [\mathcal{C}^{op}, Set]$

is represented by the object $R(d)$, and naturally so.

2. In terms of the Yoneda embedding

$y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set]$

we have

(6)$Hom_{\mathcal{C}}(-,R(d)) = y(R(d))$

The condition (2) says equivalently that $R$ has to be such that for all morphisms $h \;\colon\; d_1 \to d_2$ the following diagram in the category of presheaves $[\mathcal{C}^{op}, Set]$ commutes

$\array{ Hom_{\mathcal{D}}(L(-),d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-,R(d_1)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}( L(-) , h ) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( -, R(h) ) }} \\ Hom_{\mathcal{D}}(L(-),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-, R(d_2)) }$

This manifestly has a unique solution

$y(R(h)) \;=\; Hom_{\mathcal{C}}(-,R(h))$

for every morphism $h \colon d_1 \to d_2$ under $y(R(-))$ (6). But the Yoneda embedding $y$ is a fully faithful functor (this prop.), which means that thereby also $R(h)$ is uniquely fixed.

Remark

In more fancy language, the statement of Prop. is the following:

By precomposition $L$ defines a functor of presheaf categories

$L^* \;\colon\; [\mathcal{D}^{op}, Set] \to [\mathcal{C}^{op}, Set] \,.$

By restriction along the Yoneda embedding $y \;\colon\; \mathcal{D} \to [\mathcal{D}^{op}, Set]$ this yields the functor

$\bar L \;\colon\; \array{ \mathcal{D} &\overset{y}{\longrightarrow}& [\mathcal{D}^{op}, Set] &\overset{L^*}{\longrightarrow}& [\mathcal{C}^{op}, Set] \\ d &\mapsto& Hom_{\mathcal{D}}(-,d) &\mapsto& Hom_{\mathcal{D}}(L(-),d) } \,.$

The statement is that for all $d \in D$ this presheaf $\bar L(d)$ is representable, then it is functorially so in that there exists a functor $R \colon \mathcal{D} \to \mathcal{C}$ such that

$\bar L \;\simeq\; y \circ R \,.$

Local definition

Remark

The perspective of Prop. 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 $\mathcal{D}$:

It may happen that

$\bar L(d) \coloneqq Hom_D(L(-),d) \in [C^{op}, Set]$

is representable for some object $d \in \mathcal{D}$ but not for all $d$. The representing object may still usefully be thought of as $R(d)$, and in fact it may be viewed as a right adjoint to $L$ relative to the inclusion of the full subcategory determined by those $d$s for which $\bar L(d)$ is representable; see relative adjoint functor for more.

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

as discussed there.

In terms of universal factorization through a (co)unit

We have seen in Prop. that the unit of an adjunction and counit of an adjunction plays a special role. One may amplify this by characterizing these morphisms as universal arrows in the sense of the following Def. . In fact the existence of these is already equivalent to the existence of an adjoint functor, this is the statement of Prop. below.

$\,$

Definition

(universal arrow)

Given a functor $R \;\colon\; \mathcal{D} \to \mathcal{C}$, and an object $c\in \mathcal{C}$, a universal arrow from $c$ to $R$ is an initial object of the comma category $(c/R)$. This means that it consists of

1. an object $L(c)\in \mathcal{D}$

2. a morphism $\eta_c \;\colon\; c \to R(L(c))$, to be called the unit,

such that for any $d\in \mathcal{D}$, any morphism $f \colon c\to R(d)$ factors through this unit $\eta_c$ as

(7)$\array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \widetilde f}{\longrightarrow}&& d }$

for a unique $\widetilde f \;\colon\; L(c) \longrightarrow d$, to be called the adjunct of $f$.

Proposition

(universal morphisms are initial objects in the comma category)

Let $R: \mathcal{D} \to \mathcal{C}$ be a functor and $c \in \mathcal{C}$ an object. Then the following are equivalent:

1. $c \overset{\eta_c}{\to} R(L(c))$ is a universal morphism into $R(L(c))$ (Def. );

2. $(c, \eta_c)$ is the initial object in the comma category $c/R$.

Proposition

(collection of universal arrows equivalent to adjoint functor)

Let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a functor. Then the following are equivalent:

1. $R$ has a left adjoint functor $L \colon \mathcal{C} \to \mathcal{D}$ according to Def. ,

2. for every object $c \in \mathcal{C}$ there is a universal arrow $c \overset{\eta_c}{\longrightarrow} R(L(c))$, according to Def. .

Proof

In one direction, assume a left adjoint $L$ is given. Define the would-be universal arrow at $c \in \mathcal{C}$ to be the unit of the adjunction $\eta_c$ via Def. . Then the statement that this really is a universal arrow is implied by Prop. .

In the other direction, assume that universal arrows $\eta_c$ are given. The uniqueness clause in Def. immediately implies bijections

$\array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ \left( L(c) \overset{\widetilde f}{\to} d \right) &\mapsto& \left( c \overset{\eta_c}{\to} R(L(c)) \overset{ R(\widetilde f) }{\to} R(d) \right) }$

Hence to satisfy (1) it remains to show that these are natural in both variables. In fact, by Prop. it is sufficient to show naturality in the variable $d$. But this is immediate from the functoriality of $R$ applied in (7): For $h \colon d_1 \to d_2$ any morphism, we have

$\array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R (L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d_1) \\ && {}_{\mathllap{ R( h\circ \widetilde f ) }}\searrow && \downarrow^{\mathrlap{R(h)}} \\ && && R(d_2) }$
Example

(localization via universal arrows)

The characterization of adjoint functors in terms of universal factorizations through the unit and counit (Prop. ) is of particular interest in the case that $R$ is a full and faithful functor

$R \;\colon\; \mathcal{D} \hookrightarrow \mathcal{C}$

exhibiting $\mathcal{D}$ as a reflective subcategory of $\mathcal{C}$. In this case we may think of $L$ as a localization and of objects in the essential image of $L$ as local objects. Then the above says that:

• every morphism $c \to R d$ from $c$ into a local object factors throught the localization of $c$.

In terms of cographs/correspondences/heteromorphisms

Every profunctor

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

defines a category $C *^k D$ with $Obj(C *^k D) = Obj(C) \sqcup Obj(D)$ and with hom set given by

$Hom_{C^{op} \times D}(X,Y) = \left\{ \array{ Hom_C(X,Y) & if X, Y \in C \\ Hom_{D}(X,Y) & if X,Y \in D \\ k(X,Y) & if X \in C and Y \in D \\ \emptyset & otherwise } \right.$

($k(X,Y)$ is also called the heteromorphisms).

This category naturally comes with a functor to the interval category

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

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

$k_L(X,Y) = Hom_D(L(X), Y)$

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

$k_R(X,Y) = Hom_C(X, R(Y)) \,.$

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

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

In terms of graphs/2-sided discrete fibrations

Functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if we have a commutative diagram

$\array{ (L \downarrow Id_D) &&\stackrel{\cong}{\to}&& (Id_C \downarrow R) \\ & \searrow && \swarrow \\ && C \times D }$

where the downwards arrows are the maps induced by the projections of the comma categories. This definition of adjoint functors was introduced by Lawvere in his Ph.D. thesis, and was the original motivation for comma categories.

This diagram can be recovered directly from the image under the equivalence $[C^{op} \times D, Set] \stackrel{\simeq}{\to} DFib(D,C)$ described at 2-sided fibration of the isomorphism of induced profunctors $C^{op} \times D \to Set$ (see above at “In terms of Hom isomorphism”). Its relation to the hom-set definition of adjoint functors can thus be understood within the general paradigm of Grothendieck construction-like correspondences.

In terms of Kan extensions/liftings

Given $L \colon C \to D$, we have that it has a right adjoint $R\colon D \to C$ precisely if the left Kan extension $Lan_L 1_C$ of the identity along $L$ exists and is absolute, in which case

$R \simeq \mathop{Lan}_L 1_C \,.$

In this case, the universal 2-cell $1_C \to R L$ corresponds to the unit of the adjunction; the counit and the verification of the triangular identities can all be obtained through properties of Kan extensions and absoluteness.

It is also possible to express this in terms of Kan liftings: $L$ has a right adjoint $R$ if and only if:

• $R \simeq \mathop{Rift}_L 1_D$ and this Kan lift is absolute

In this case, we get the counit as given by the universal cell $L R \to 1_D$, while the rest of the data and properties can be derived from it through the absolute Kan lifting assumption.

Dually, we have that for $R\colon D \to C$, it has a left adjoint $L \colon C \to D$ precisely if

• $L \simeq \mathop{Ran}_R 1_D$, and this Kan extension is absolute

or, in terms of left Kan liftings:

• $L \simeq \mathop{Lift}_R 1_C$, and this Kan lifting is absolute

The formulations in terms of liftings generalize to relative adjoints by allowing an arbitrary functor $J$ in place of the identity; see there for more.

Properties

Basic properties

Proposition

(adjoint functors are unique up to natural isomorphism)

The left adjoint or right adjoint to a functor (Def. ), if it exists, is unique up to natural isomorphism.

Proof

Suppose the functor $L \colon \mathcal{D} \to \mathcal{C}$ is given, and we are asking for uniqueness of its right adjoint, if it exists. The other case is directly analogous.

Suppose that $R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D}$ are two functors which are right adjoint to $L$. Then for each $d \in \mathcal{D}$ the corresponding two hom-isomorphisms (1) combine to say that there is a natural isomorphism

$\Phi_d \;\colon\; Hom_{\mathcal{C}}(-,R_1(d)) \;\simeq\; Hom_{\mathcal{C}}(-,R_2(d))$

As in the proof of Prop. , the Yoneda lemma implies that

$\Phi_d \;=\; y( \phi_d )$

for some isomorphism

$\phi_d \;\colon\; R_1(d) \overset{\simeq}{\to} R_2(d) \,.$

But then the uniqueness statement of Prop. implies that the collection of these isomorphisms for each object constitues a natural isomorphism between the functors.

Proposition

Let $(L \dashv R) \colon \mathcal{D} \to \mathcal{C}$ be a pair of adjoint functors (Def. ). Then

• $L$ preserves all colimits that exist in $\mathcal{C}$,

• $R$ preserves all limits in $\mathcal{D}$.

Proof

Let $y : I \to \mathcal{D}$ be a diagram whose limit $\lim_{\leftarrow_i} y_i$ exists. Then we have a sequence of natural isomorphisms, natural in $x \in C$

\begin{aligned} Hom_{\mathcal{C}}(x, R {\lim_\leftarrow}_i y_i) & \simeq Hom_{\mathcal{D}}(L x, {\lim_\leftarrow}_i y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{D}}(L x, y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{C}}( x, R y_i) \\ & \simeq Hom_{\mathcal{C}}( x, {\lim_\leftarrow}_i R y_i) \,, \end{aligned}

where we used the hom-isomorphism (1) and the fact that any hom-functor preserves limits (see there). Because this is natural in $x$ the Yoneda lemma implies that we have an isomorphism

$R {\lim_\leftarrow}_i y_i \simeq {\lim_\leftarrow}_i R y_i \,.$

The argument that shows the preservation of colimits by $L$ is analogous.

Remark

A partial converse to Prop. is provided by the adjoint functor theorem. See also Pointwise Expression below.

Proposition

Let $L \dashv R$ be a pair of adjoint functors (Def. ). Then the following holds:

• $R$ is faithful precisely if the component of the counit over every object $x$ is an epimorphism $L R x \stackrel{}{\to} x$;

• $R$ is full precisely if the component of the counit over every object $x$ is a split monomorphism $L R x \stackrel{}{\to} x$;

• $L$ is faithful precisely if the component of the unit over every object $x$ is a monomorphism $x \hookrightarrow R L x$;

• $L$ is full precisely if the component of the unit over every object $x$ is a split epimorphism $x \to R L x$;

• $R$ is full and faithful (exhibits a reflective subcategory) precisely if the counit is a natural isomorphism $\epsilon : L \circ R \stackrel{\simeq}{\to} Id_D$

• $L$ is full and faithful (exhibits a coreflective subcategory) precisely if the unit is a natural isomorphism $\eta : Id_C \stackrel{\simeq}{\to} R \circ L$.

• The following are equivalent:

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

• $L$ is an equivalence;

• $R$ is an equivalence.

$\phantom{A}$adjunction$\phantom{A}$$\phantom{A}$unit is iso:$\phantom{A}$
$\phantom{A}$coreflection$\phantom{A}$
$\phantom{A}$counit is iso:$\phantom{A}$$\phantom{A}$reflection$\phantom{A}$$\phantom{A}$adjoint equivalence$\phantom{A}$
Proof

For the characterization of faithful $R$ by epi counit components, notice (as discussed at epimorphism ) that $L R x \to x$ being an epimorphism is equivalent to the induced function

$Hom(x, a) \to Hom(L R x, a)$

being an injection for all objects $a$. Then use that, by adjointness, we have an isomorphism

$Hom(L R x , a ) \stackrel{\simeq}{\to} Hom(R x, R a)$

and that, by the formula for adjuncts and the zig-zag identity, this is such that the composite

$R_{x,a} : Hom(x,a) \to Hom(L R x, a) \stackrel{\simeq}{\to} Hom(R x, R a)$

is the component map of the functor $R$ (this Prop.):

\begin{aligned} (x \stackrel{f}{\to} a) & \mapsto (L R x \to x \stackrel{f}{\to} a) \\ & \mapsto (R L R x \to R x \stackrel{R f}{\to} R a) \\ & \mapsto (R x \to R L R x \to R x \stackrel{R f}{\to} R a) \\ & = (R x \stackrel{R f}{\to} R a) \end{aligned} \,.

Therefore $R_{x,a}$ is injective for all $x,a$, hence $R$ is faithful, precisely if $L R x \to x$ is an epimorphism for all $x$. The characterization of $R$ full is just the same reasoning applied to the fact that $\epsilon_x \colon L R x \to x$ is a split monomorphism iff for all objects $a$ the induced function

$Hom(x, a) \to Hom(L R x, a)$

is a surjection.

For the characterization of faithful $L$ by monic units notice that analogously (as discussed at monomorphism) $x \to R L x$ is a monomorphism if for all objects $a$ the function

$Hom(a,x ) \to Hom(a, R L x)$

is an injection. Analogously to the previous argument we find that this is equivalent to

$L_{a,x} : Hom(a,x ) \to Hom(a, R L x) \stackrel{\simeq}{\to} Hom(L a, L x)$

being an injection. So $L$ is faithful precisely if all $x \to R L x$ are monos. For $L$ full, it’s just the same applied to $x \to R L x$ split epimorphism iff the induced function

$Hom(a,x ) \to Hom(a, R L x)$

is a surjection, for all objects $a$.

The proof of the other statements proceeds analogously.

Parts of this statement can be strengthened:

Proposition

Let $(L \dashv R) : D \to C$ be a pair of adjoint functors such that there is any natural isomorphism

$L R \simeq Id \,,$

then also the counit $\epsilon : L R \to Id$ is an isomorphism.

This appears as (Johnstone, lemma 1.1.1).

Proof

Using the given isomorphism, we may transfer the comonad structure on $L R$ to a comonad structure on $Id_D$. By the Eckmann-Hilton argument the endomorphism monoid of $Id_D$ is commutative. Therefore, since the coproduct on the comonad $Id_D$ is a left inverse to the counit (by the co-unitality property applied to this degenerate situation), it is in fact a two-sided inverse and hence the $Id_D$-counit is an isomorphism. Transferring this back one finds that also the counit of the comand $L R$, hence of the adjunction $(L \dashv R)$ is an isomorphism.

Pointwise expression

Proposition

(pointwise expression of left adjoints in terms of limits over comma categories)

A functor $R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ has a left adjoint $L \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ precisely if

1. $R$ preserves all limits that exist in $\mathcal{C}$;

2. for each object $d \in \mathcal{D}$, the limit of the canonical functor out of the comma category of $R$ under $d$

$d/R \longrightarrow \mathcal{C}$

exists.

In this case the value of the left adjoint $L$ on $d$ is given by that limit:

(8)$L(d) \;\simeq\; \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c$
Proof

First assume that the left adjoint exist. Then

1. $R$ is a right adjoint and hence preserves limits since all right adjoints preserve limits;

2. by Prop. the adjunction unit provides a universal morphism $\eta_d$ into $L(d)$, and hence, by Prop. , exhibits $(L(d), \eta_d)$ as the initial object of the comma category $d/R$. The limit over any category with an initial object exists, as it is given by that initial object.

Conversely, assume that the two conditions are satisfied and let $L(d)$ be given by (8). We need to show that this yields a left adjoint.

By the assumption that $R$ preserves all limits that exist, we have

(9)$\array{ R(L(d)) & = R\left( \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c \right) \\ & \simeq \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} R(c) }$

Since the $d \overset{f}{\to} R(d)$ constitute a cone over the diagram of the $R(d)$, there is universal morphism

$d \overset{\phantom{AA} \eta_d \phantom{AA}}{\longrightarrow} R(L(d)) \,.$

By Prop. it is now sufficient to show that $\eta_d$ is a universal morphism into $L(d)$, hence that for all $c \in \mathcal{C}$ and $d \overset{g}{\longrightarrow} R(c)$ there is a unique morphism $L(d) \overset{\widetilde f}{\longrightarrow} c$ such that

$\array{ && d \\ & {}^{\mathllap{ \eta_d }}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(d)) && \underset{\phantom{AA}R(\widetilde f)\phantom{AA}}{\longrightarrow} && R(c) \\ L(d) &&\underset{\phantom{AA}\widetilde f\phantom{AA}}{\longrightarrow}&& c }$

By Prop. , this is equivalent to $(L(d), \eta_d)$ being the initial object in the comma category $c/R$, which in turn is equivalent to it being the limit of the identity functor on $c/R$ (this prop.). But this follows directly from the limit formulas (8) and (9).

See at adjoint functor theorem for more.

Every adjunction $(L \dashv R)$ induces a monad $R \circ L$ and a comonad $L \circ R$. There is in general more than one adjunction which gives rise to a given monad this way, in fact there is a category of adjunctions for a given monad. The initial object in that category is the adjunction over the Kleisli category of the monad and the terminal object is that over the Eilenberg-Moore category of algebras. (e.g. Borceux, Vol. 2, Proposition 4.2.1) The latter is called the monadic adjunction.

Given a pair of adjoint functors

$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{C}$

there is induced an opposite adjunction of opposite functors between their opposite categories of the form

$\mathcal{D}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{C}^{op} \,.$

Hence where $L$ was the left adjoint, its opposite becomes the right adjoint, and dually for $R$.

This is immediate from the definition of opposite categories and the characterization of adjoint functors via the corresponding hom-isomorphism.

The adjunction unit of the opposite adjunction has as components the components of the original adjunction counit, regarded in the opposite category, and dually:

$\epsilon^{R^{op} C^{op}}_{d} \;\colon\; R^{op}\circ L^{op}(d) \xrightarrow{\;\; \big( \eta^{R L}_d \big)^{op} \;\;} d \,, {\phantom{AAAAAA}} \eta^{L^{op} R^{op}}_{c} \;\colon\; c \xrightarrow{\;\; \big( \epsilon^{L R}_c \big)^{op} \;\;} L^{op} \circ R^{op}(c) \,.$

Proposition

Let

$\mathcal{D} \underoverset {\underset{\;\;\;\;R\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}$

be a pair of adjoint functors (adjoint ∞-functors), where the category (∞-category) $\mathcal{C}$ has all pullbacks (homotopy pullbacks).

Then:

1. For every object $b \in \mathcal{C}$ there is induced a pair of adjoint functors between the slice categories (slice ∞-categories) of the form

(10)$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/b} \mathrlap{\,,}$

where:

• $L_{/b}$ is the evident induced functor (applying $L$ to the entire triangle diagrams in $\mathcal{C}$ which represent the morphisms in $\mathcal{C}_{/b}$);

• $R_{/b}$ is the composite

$R_{/b} \;\colon\; \mathcal{D}_{/{L(b)}} \overset{\;\;R\;\;}{\longrightarrow} \mathcal{C}_{/{(R \circ L(b))}} \overset{\;\;(\eta_{b})^*\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $R$;

2. the (homotopy) pullback along the $(L \dashv R)$-unit at $b$ (i.e. the base change along $\eta_b$).

2. For every object $b \in \mathcal{D}$ there is induced a pair of adjoint functors between the slice categories of the form

(11)$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot} \mathcal{C}_{/R(b)} \mathrlap{\,,}$

where:

• $R_{/b}$ is the evident induced functor (applying $R$ to the entire triangle diagrams in $\mathcal{D}$ which represent the morphisms in $\mathcal{D}_{/b}$);

• $L_{/b}$ is the composite

$L_{/b} \;\colon\; \mathcal{D}_{/{R(b)}} \overset{\;\;L\;\;}{\longrightarrow} \mathcal{C}_{/{(L \circ R(b))}} \overset{\;\;(\epsilon_{b})_!\;\;}{\longrightarrow} \mathcal{C}_{/b}$

of

1. the evident functor induced by $L$;

2. the composition with the $(L \dashv R)$-counit at $b$ (i.e. the left base change along $\epsilon_b$).

The first statement appears, in the generality of (∞,1)-category theory, as HTT, prop. 5.2.5.1. For discussion in model category theory see at sliced Quillen adjunctions.
Proof

Recall that (this Prop.) the hom-isomorphism that defines an adjunction of functors (this Def.) is equivalently given in terms of composition with

• the adjunction unit $\;\;\eta_c \colon c \xrightarrow{\;} R \circ L(c)$

• the adjunction counit $\;\;\epsilon_d \colon L \circ R(d) \xrightarrow{\;} d$

as follows:

Using this, consider the following transformations of morphisms in slice categories, for the first case:

(1a)

(2a)

(2b)

(1b)

Here:

• (1a) and (1b) are equivalent expressions of the same morphism $f$ in $\mathcal{D}_{/L(b)}$, by (at the top of the diagrams) the above expression of adjuncts between $\mathcal{C}$ and $\mathcal{D}$ and (at the bottom) by the triangle identity.

• (2a) and (2b) are equivalent expression of the same morphism $\tilde f$ in $\mathcal{C}_{/b}$, by the universal property of the pullback.

Hence:

• starting with a morphism as in (1a) and transforming it to $(2)$ and then to (1b) is the identity operation;

• starting with a morphism as in (2b) and transforming it to (1) and then to (2a) is the identity operation.

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the first claimed form (10).

The second case follows analogously, but a little more directly since no pullback is involved:

(1a)

(2)

(1b)

In conclusion, the transformations (1) $\leftrightarrow$ (2) consitute a hom-isomorphism that witnesses an adjunction of the second claimed form (11).

Remark

The sliced adjunction (Prop. ) in the second form (11) is such that the sliced left adjoint sends slicing morphism $\tau$ to their adjuncts $\widetilde{\tau}$, in that (again by this Prop.):

$L_{/d} \, \left( \array{ c \\ \big\downarrow {}^{\mathrlap{\tau}} \\ R(b) } \right) \;\; = \;\; \left( \array{ L(c) \\ \big\downarrow {}^{\mathrlap{\widetilde{\tau}}} \\ b } \right) \;\;\; \in \; \mathcal{D}_{/b}$

Examples

To a fair extent, category theory is all about adjoint functors and the other universal constructions: Kan extensions, limits, representable functors, which are all special cases of adjoint functors – and adjoint functors are special cases of these.

Listing examples of adjoint functors is much like listing examples of integrals in analysis: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see coend for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.

General

• A pair of adjoint functors between posets is a Galois correspondence.

• A pair of adjoint functors $(L \dashv R)$ where $R$ is a full and faithful functor exhibits a reflective subcategory.

In this case $L$ may be regarded as a localization. The fact that the adjunction provides universal factorization through unit and counit in this case means that every morphism $f : c \to R d$ into a local object factors through the localization of $c$.

• A pair of adjoint functors that is also an equivalence of categories is called an adjoint equivalence.

• A pair of adjoint functors where $C$ and $D$ have finite limits and $L$ preserves these finite limits is a geometric morphism. These are one kind of morphisms between toposes. If in addition $R$ is full and faithful, then this is a geometric embedding.

• The left and right adjoint functors $p_!$ and $p_*$ (if they exist) to a functor $p^* : [K',C] \to [K,C]$ between functor categories obtained by precomposition with a functor $p : K \to K'$ of diagram categories are called the left and right Kan extension functors along $p$

$(Lan_p \dashv p^* \dashv Ran_p) := (p_! \dashv p^* \dashv p_*) : [K,C] \stackrel{\overset{p_!}{\to}}{\stackrel{\overset{p^*}{\leftarrow}}{\underset{p_*}{\to}}} [K',C] \,.$

If $K' = {*}$ is the terminal category then this are the limit and colimit functors on $[K,C]$.

If $C =$ Set then this is the direct image and inverse image operation on presheaves.

• if $R$ is regarded as a forgetful functor then its left adjoint $L$ is a regarded as a free functor.

• If $C$ is a category with small colimits and $K$ is a small category (a diagram category) and $Q : K \to C$ is any functor, then this induces a nerve and realization pair of adjoint functors

$(|-|_Q \dashv N_Q) : C \stackrel{\overset{|-|_Q}{\leftarrow}}{\underset{N_Q}{\to}} [K^{op}, Set]$

between $C$ and the category of presheaves on $K$, where

• the nerve functor is given by

$N_Q(c) := Hom_C(Q(-),c) : k \mapsto Hom_C(Q(k),c)$
• and the realization functor is given by the coend

$|F|_Q := \int^{k \in K} Q(k)\cdot F(k) \,,$

where in the integrand we have the canonical tensoring of $C$ over Set ($Q(k) \cdot F(k) = \coprod_{s \in F(k)} Q(k)$).

A famous examples of this is obtained for $C =$ Top, $K = \Delta$ the simplex category and $Q : \Delta \to Top$ the functor that sends $[n]$ to the standard topological $n$-simplex. In this case the nerve functor is the singular simplicial complex functor and the realization is ordinary geometric realization.

Related concepts

For the basics, see any text on category theory (and see the references at adjunction), for instance:

Though the definition of an adjoint equivalence appears in Grothendieck's Tohoku paper, the idea of adjoint functors in general goes back to

• Daniel Kan, Adjoint functors, Transactions of the American Mathematical Society Vol. 87, No. 2 (Mar., 1958), pp. 294-329 (jstor)

and its fundamental relevance for category theory was realized due to

• Peter Freyd, Abelian categories – An introduction to the theory of functors, Harper’s Series in Modern Mathematics, Harper & Row, New York, 1964 (pdf).

• William Lawvere, Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296

The history of the idea that adjoint functors formalize aspects of dialectics is recounted in

• Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) (doi:10.1007/978-94-009-8462-2_6)

(more along these lines at adjoint modality)