nLab
adjoint functor

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 LRL \dashv R in terms of a natural bijection between hom-sets (Def. below):

{L(c)d}{cR(d)} \{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.

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Definition

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

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

𝒟RL𝒞 \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 LL left adjoint and RR right adjoint, denoted

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

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

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

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

Hom 𝒟(L(c),d) Hom 𝒞(c,R(d)) (L(c)fd) (cf˜R(d)) \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 cc and dd. This isomorphism is the adjunction isomorphism and the image f˜\widetilde f of a morphism ff under this bijections is called the adjunct of ff. Conversely, ff is called the adjunct of f˜\widetilde f.

Naturality here means that for every morphism g:c 2c 1g \colon c_2 \to c_1 in 𝒞\mathcal{C} and for every morphisms h:d 1d 2h\colon d_1\to d_2 in 𝒟\mathcal{D}, the resulting square

(2)Hom 𝒟(L(c 1),d 1) ()˜ Hom 𝒞(c 1,R(d 1)) Hom 𝒟(L(g),h) Hom 𝒞(g,R(h)) Hom 𝒟(L(c 2),d 2) ()˜ Hom 𝒞(c 2,R(d 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:L(c 1)d 1f \;\colon\; L(c_1) \to d_1 with adjunct f˜:c 1R(d 1)\widetilde f \;\colon\; c_1 \to R(d_1), the adjunct of the composition is

L(c 1) f d 1 L(g) h L(c 2) d 2˜=c 1 f˜ R(d 1) g R(h) c 2 R(d 2) \widetilde{ \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

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

according to Def. one says that

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

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

    ϵ d:L(R(d))d \epsilon_d \;\colon\; L(R(d)) \longrightarrow d
Proposition

(general adjuncts in terms of unit/counit)

Consider a pair of adjoint functors

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

according to Def. , with adjunction units η c\eta_c and adjunction counits ϵ d\epsilon_d according to Def. .

Then

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

    (3)f˜:cη cR(L(c))R(f)R(d) \widetilde f \;\colon\; c \overset{\eta_c}{\longrightarrow} R(L(c)) \overset{R(f)}{\longrightarrow} R(d)

    Conversely, the adjunct ff of any morphism cf˜R(d)c \overset{\widetilde f}{\longrightarrow} R(d) is obtained from LL and ϵ d\epsilon_d as

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

    η:Id 𝒞RL \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L

    and

    ϵ:LRId 𝒟 \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):L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) 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):R(d)η R(d)R(L(R(d)))R(ϵ d)R(d) 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

id L(c) Hom 𝒟(L(c),L(c)) ()˜ Hom 𝒞(c,R(L(c))) Hom 𝒟(L(id),f) Hom 𝒞(id,R(f)) Hom 𝒟(L(c),d) ()˜ Hom 𝒞(c,R(d)) \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)id_{L(c_1)} in the top left entry. Its image under going down and then right in the diagram is f˜\widetilde f, by Def. . On the other hand, its image under going right and then down is R(f)η c 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 ff.

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:

id L(c) =id L(c)˜˜ =cη cR(L(c))˜ =L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) \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 moprhism f:c 1c 2f \colon c_1 \to c_2 the following square commutes:

c 1 f c 2 η c 1 η c 2 R(L(c 1)) R(L(f)) R(L(c 2)) \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

id L(c 2) Hom 𝒟(L(c 2),L(c 2)) ()˜ Hom 𝒞(c 2,R(L(c 2))) Hom 𝒟(L(f),id L(c 2)) Hom 𝒞(f,R(id L(c 2))) Hom 𝒟(L(c 1),L(c 2)) ()˜ Hom 𝒞(c 1,R(L(c 1))) \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_1))) }

The image of the element id L(c 2)id_{L(c_2)} in the top left along the right and down is fη c 2 f \circ \eta_{c_2}, by Def. , while its image down and then to the right is L(f)˜=R(L(f))η c 1\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 ϵ\epsilon is directly analogous.

Proposition

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

Two functors

𝒟RL𝒞 \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

    η:Id 𝒞RL \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L

    and

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

    id L(c):L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) 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):R(d)η R(d)R(L(R(d)))R(ϵ d)R(d) 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 ff˜f \mapsto \widetilde f is an isomorphism follows from the computation

f˜˜ =cη cR(L(c))R(f)R(d)˜ =L(c)L(η c)L(R(L(c)))L(R(f))L(R(d))ϵ dd =L(c)L(η c)L(R(L(c)))ϵ L(c)L(c)fd =L(c)fd \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 ϵ\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:

c 2 η c 2 R(L(c 2)) g R(L(g)) R(L(g)f) c 1 η c 1 R(L(c 1)) R(f) R(d 1) R(hf) R(h) R(d 2) \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 LRL \dashv R in Def. implies in particular that for every object d𝒟d \in \mathcal{D} the functor Hom 𝒟(L(),d)Hom_{\mathcal{D}}(L(-),d) is a representable functor with representing object R(d)R(d). The following Prop. observes that the existence of such representing objects for all dd 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:𝒞𝒟L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} has a right adjoint R:𝒟𝒞R \;\colon\; \mathcal{D} \to \mathcal{C}, according to Def. , already if for all objects d𝒟d \in \mathcal{D} there is an object R(d)𝒞R(d) \in \mathcal{C} such that there is a natural isomorphism

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

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

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

such that for each morphism g:c 2c 1g \;\colon\; c_2 \to c_1, the following diagram commutes

(5)Hom 𝒟(L(c 1),d) ()˜ Hom 𝒞(c 1,R(d)) Hom 𝒞(L(g),id d) Hom 𝒞(f,id R(d)) Hom 𝒟(L(c 2),d) ()˜ Hom 𝒞(c 2,R(d)) \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 RR from a function on objects to a function on morphisms such as to make it a functor R:𝒟𝒞R \colon \mathcal{D} \to \mathcal{C} which is right adjoint to LL. , 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 d𝒟d \in \mathcal{D} the presheaf

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

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

  2. In terms of the Yoneda embedding

    y:𝒟[𝒟 op,Set] y \;\colon\; \mathcal{D} \hookrightarrow [\mathcal{D}^{op}, Set]

    we have

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

The condition (2) says equivalently that RR has to be such that for all morphisms h:d 1d 2h \;\colon\; d_1 \to d_2 the following diagram in the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] commutes

Hom 𝒟(L(),d 1) ()˜ Hom 𝒞(,R(d 1)) Hom 𝒞(L(),h) Hom 𝒞(,R(h)) Hom 𝒟(L(),d 2) ()˜ Hom 𝒞(,R(d 2)) \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 𝒞(,R(h)) y(R(h)) \;=\; Hom_{\mathcal{C}}(-,R(h))

for every morphism h:d 1d 2h \colon d_1 \to d_2 under y(R())y(R(-)) (6). But the Yoneda embedding yy is a fully faithful functor (this prop.), which means that thereby also R(h)R(h) is uniquely fixed.

Remark

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

By precomposition LL defines a functor of presheaf categories

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

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

L¯:𝒟 y [𝒟 op,Set] L * [𝒞 op,Set] d Hom 𝒟(,d) Hom 𝒟(L(),d). \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 dDd \in D this presheaf L¯(d)\bar L(d) is representable, then it is functorially so in that there exists a functor R:𝒟𝒞R \colon \mathcal{D} \to \mathcal{C} such that

L¯yR. \bar L \;\simeq\; y \circ R \,.

Local definition

Remark

(relative adjoint functors)

The perspective of Prop. has the advantage that it yields useful information even if the adjoint functor RR does not exist globally, i.e. as a functor on all of 𝒟\mathcal{D}:

It may happen that

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

is representable for some object d𝒟d \in \mathcal{D} but not for all dd. The representing object may still usefully be thought of as R(d)R(d), and in fact it may be viewed as a right adjoint to LL relative to the inclusion of the full subcategory determined by those dds for which L¯(d)\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.

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Definition

(universal arrow)

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

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

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

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

(7) c η c f R(L(c)) R(f˜) R(d) L(c) f˜ d \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 f˜:L(c)d\widetilde f \;\colon\; L(c) \longrightarrow d, to be called the adjunct of ff.

(e.g Borceux, vol 1, Def. 3.1.1)

Proposition

(universal morphisms are initial objects in the comma category)

Let 𝒞R𝒟\mathcal{C} \overset{R}{\longrightarrow} \mathcal{D} be a functor and d𝒟d \in \mathcal{D} an object. Then the following are equivalent:

  1. dη dR(c)d \overset{\eta_d}{\to} R(c) is a universal morphism into R(c)R(c) (Def. );

  2. (d,η d)(d, \eta_d) is the initial object in the comma category d/Rd/R.

Proposition

(collection of universal arrows equivalent to adjoint functor)

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

  1. RR has a left adjoint functor L:𝒞𝒟L \colon \mathcal{C} \to \mathcal{D} according to Def. ,

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

Proof

In one direction, assume a left adjoint LL is given. Define the would-be universal arrow at c𝒞c \in \mathcal{C} to be the unit of the adjunction η c\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 η c\eta_c are given. The uniqueness clause in Def. immediately implies bijections

Hom 𝒟(L(c),d) Hom 𝒞(c,R(d)) (L(c)f˜d) (cη cR(L(c))R(f˜)R(d)) \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 dd. But this is immediate from the functoriality of RR applied in (7): For h:d 1d 2h \colon d_1 \to d_2 any morphism, we have

c η c f R(L(c)) R(f˜) R(d 1) R(hf˜) R(h) R(d 2) \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 RR is a full and faithful functor

R:𝒟𝒞 R \;\colon\; \mathcal{D} \hookrightarrow \mathcal{C}

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

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

In terms of cographs/correspondences/heteromorphisms

Every profunctor

k:C op×DS k : C^{op} \times D \to S

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

Hom C op×D(X,Y)={Hom C(X,Y) ifX,YC Hom D(X,Y) ifX,YD k(X,Y) ifXCandYD otherwise 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)k(X,Y) is also called the heteromorphisms).

This category naturally comes with a functor to the interval category

C* kDΔ 1. C *^k D \to \Delta^1 \,.

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

k L(X,Y)=Hom D(L(X),Y) k_L(X,Y) = Hom_D(L(X), Y)

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

k R(X,Y)=Hom C(X,R(Y)). k_R(X,Y) = Hom_C(X, R(Y)) \,.

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

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

In terms of graphs/2-sided discrete fibrations

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

(LId D) (Id CR) C×D \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×D,Set]DFib(D,C)[C^{op} \times D, Set] \stackrel{\simeq}{\to} DFib(D,C) described at 2-sided fibration of the isomorphism of induced profunctors C op×DSetC^{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:CDL \colon C \to D, we have that it has a right adjoint R:DCR\colon D \to C precisely if the left Kan extension Lan L1 CLan_L 1_C of the identity along LL exists and is absolute, in which case

RLan L1 C. R \simeq \mathop{Lan}_L 1_C \,.

In this case, the universal 2-cell 1 CRL1_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: LL has a right adjoint RR if and only if:

In this case, we get the counit as given by the universal cell LR1 DL 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:DCR\colon D \to C, it has a left adjoint L:CDL \colon C \to D precisely if

  • LRan R1 DL \simeq \mathop{Ran}_R 1_D, and this Kan extension is absolute

or, in terms of left Kan liftings:

  • LLift R1 CL \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 JJ 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:𝒟𝒞L \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:𝒞𝒟R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D} are two functors which are right adjoint to LL. Then for each d𝒟d \in \mathcal{D} the corresponding two hom-isomorphisms (1) combine to say that there is a natural isomorphism

Φ d:Hom 𝒞(,R 1(d))Hom 𝒞(,R 2(d)) \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

Φ d=y(ϕ d) \Phi_d \;=\; y( \phi_d )

for some isomorphism

ϕ d:R 1(d)R 2(d). \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

(left adjoints preserve colimits and right adjoints preserve limits)

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

Proof

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

Hom 𝒞(x,Rlim iy i) Hom 𝒟(Lx,lim iy i) lim iHom 𝒟(Lx,y i) lim iHom 𝒞(x,Ry i) Hom 𝒞(x,lim iRy i), \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 xx the Yoneda lemma implies that we have an isomorphism

Rlim iy ilim iRy i. R {\lim_\leftarrow}_i y_i \simeq {\lim_\leftarrow}_i R y_i \,.

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

Remark

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

Proposition

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

A\phantom{A}adjunctionA\phantom{A}A\phantom{A}unit is iso:A\phantom{A}
A\phantom{A}coreflectionA\phantom{A}
A\phantom{A}counit is iso:A\phantom{A}A\phantom{A}reflectionA\phantom{A}A\phantom{A}adjoint equivalenceA\phantom{A}
Proof

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

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

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

Hom(LRx,a)Hom(Rx,Ra) 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)Hom(LRx,a)Hom(Rx,Ra) 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 RR:

(xfa) (LRxxfa) (RLRxRxRfRa) (RxRLRxRxRfRa) =(RxRfRa). \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,aR_{x,a} is injective for all x,ax,a, hence RR is faithful, precisely if LRxxL R x \to x is an epimorphism for all xx. The characterization of RR full is just the same reasoning applied to the fact that ϵ x:LRxx\epsilon_x \colon L R x \to x is a split monomorphism iff for all objects aa the induced function

(8)Hom(x,a)Hom(LRx,a) Hom(x, a) \to Hom(L R x, a)

is a surjection.

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

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

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

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

is a surjection, for all objects aa.

The proof of the other statements proceeds analogously.

Parts of this statement can be strenghened:

Proposition

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

LRId, L R \simeq Id \,,

then also the counit ϵ:LRId\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 LRL R to a comonad structure on Id DId_D. By the Eckmann-Hilton argument the endomorphism monoid of Id DId_D is commutative. Therefore, since the coproduct on the comonad Id DId_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 DId_D-counit is an isomorphism. Transferring this back one finds that also the counit of the comand LRL R, hence of the adjunction (LR)(L \dashv R) is an isomorphism.

Pointwise expression

Proposition

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

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

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

  2. for each object d𝒟d \in \mathcal{D}, the limit of the canonical functor out of the comma category of RR under dd

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

    exists.

In this case the value of the left adjoint LL on dd is given by that limit:

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

(e.g. MacLane, chapter X, theorem 2)

Proof

First assume that the left adjoint exist. Then

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

  2. by Prop. the adjunction unit provides a universal morphism η d\eta_d into L(d)L(d), and hence, by Prop. , exhibits (L(d),η d)(L(d), \eta_d) as the initial object of the comma category d/Rd/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)L(d) be given by (9). We need to show that this yields a left adjoint.

By the assumption that RR preserves all limits that exist, we have

(10)R(L(d)) =R(lim(c,d f R(c))d/Rc) lim(c,d f R(c))d/RR(c) \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 dfR(d)d \overset{f}{\to} R(d) constitute a cone over the diagram of the R(d)R(d), there is universal morphism

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

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

d η d f R(L(d)) AAR(f˜)AA R(c) L(d) AAf˜AA c \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),η d)(L(d), \eta_d) being the initial object in the comma category c/Rc/R, which in turn is equivalent to it being the limit of the identity functor on c/Rc/R (this prop.). But this follows directly from the limit formulas (9) and (10).

See at adjoint functor theorem for more.

Relation to monads

Every adjunction (LR)(L \dashv R) induces a monad RLR \circ L and a comonad LRL \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. prop. 4.2.2) The latter is called the monadic adjunction.

Moreover, passing from adjunctions to monads and back to their monadic adjunctions constitutes itself an adjunction between adjunctions and monads, called the semantics-structure adjunction.

Examples

The central point about examples of adjoint functors is:

Adjoint functors are ubiquitous .

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 (LR)(L \dashv R) where RR is a full and faithful functor exhibits a reflective subcategory.

    In this case LL 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:cRdf : c \to R d into a local object factors through the localization of cc.

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

  • A pair of adjoint functors where CC and DD have finite limits and LL preserves these finite limits is a geometric morphism. These are one kind of morphisms between toposes. If in addition RR is full and faithful, then this is a geometric embedding.

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

    (Lan pp *Ran p):=(p !p *p *):[K,C]p *p *p ![K,C]. (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=*K' = {*} is the terminal category then this are the limit and colimit functors on [K,C][K,C].

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

  • if RR is regarded as a forgetful functor then its left adjoint LL is a regarded as a free functor.

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

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

    between CC and the category of presheaves on KK, where

    • the nerve functor is given by

      N Q(c):=Hom C(Q(),c):kHom C(Q(k),c) 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:= kKQ(k)F(k), |F|_Q := \int^{k \in K} Q(k)\cdot F(k) \,,

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

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

References

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

For other textbook reference see any of the references listed at category theory, for instance

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)

For more on this see at adjoint modality.

See also



  1. “the universality of the concept of adjointness, which was first isolated and named in the conceptual sphere of category theory” (Lawvere 69) “In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role.” (first line from An interview with William Lawvere, paraphrasing the first paragraph of Taking categories seriously)

Last revised on July 27, 2018 at 14:09:21. See the history of this page for a list of all contributions to it.