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adjoint functor

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Definition

We say that two functors L:CDL:C\to D and R:DCR:D\to C are adjoint if they form an adjunction LRL \dashv R in the 2-category Cat of categories. This means that they are equipped with natural transformations η:1 CRL\eta: 1_C \to R \circ L and ϵ:LR1 D\epsilon: L \circ R \to 1_D satisfying the triangle identities, that is the compositions LLηLRLϵLLL \stackrel{L\eta}\to L R L\stackrel{\epsilon L}\to L and RηRRLRRϵRR\stackrel{\eta R}\to R L R \stackrel{R\epsilon}\to R are identities. The left or right adjoint of any functor, if it exists, is unique up to unique isomorphism.

We say that LL is the left adjoint of RR and that RR is the right adjoint of LL.

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

In terms of Hom isomorphism

An adjunction LRL\dashv R is equivalently given by a natural isomorphism of hom-functors C op×DSetC^{op} \times D \to Set

Hom D(L(),)Hom C(,R()). Hom_D(L(-),-) \simeq Hom_C(-,R(-)) \,.

In other words, for all cCc \in C and dDd \in D there is a bijection of sets

Hom D(L(c),d)Hom C(c,R(d)) Hom_D(L(c),d) \simeq Hom_C(c,R(d))

naturally in cc and dd. This isomorphism is the adjunction isomorphism and the image of an element under this isomorphism is its adjunct.

Given such an adjunction isomorphism, η\eta and ϵ\epsilon 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)df:L(c)\to d is R(f)ηR(f) \eta, and the adjunct of g:cR(d)g:c \to R(d) is ϵL(g)\epsilon L(g). The triangle identities are precisely what is necessary to ensure that this is an isomorphism.

In terms of representable functors

Global definition

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

Hom D(L(),d)Hom C(,R(d)).Hom_D(L(-),d) \cong Hom_C(-,R(d)).

There is then a unique way to define RR on arrows so as to make these isomorphisms natural in dd as well.

In more fancy language, by precomposition LL defines a functor

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

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

L¯:DY[D op,Set]L *[C op,Set]. \bar L : D \stackrel{Y}{\to} [D^{op}, Set] \stackrel{L^*}{\to} [C^{op}, Set] \,.

such that

L¯:dHom D(L(),d)[C op,Set]. \bar L : d \mapsto Hom_D(L(-),d) \in [C^{op}, Set] \,.

If 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 : D \to C such that

L¯YR. \bar L \simeq Y \circ R \,.

Local definition

This definition 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 DD:

it may happen that

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

is representable for some dd but not for all dd. The representing object may still usefully be thought of as R(d)R(d), and in fact it can 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 arrows / universal factorization through unit and counit

Given R:DCR : D\to C, and cCc\in C, a universal arrow from cc to RR is an initial object of the comma category (c/R)(c/R). That is, it consists of an object L(c)DL(c)\in D and a morphism i c:cR(L(c))i_c : c\to R(L(c)) – the unit – such that for any dDd\in D, any morphism f:cR(d)f : c\to R(d) factors through the unit i ci_c as

c i c f RLc Rf˜ Rd Lc f˜ d \array{ && c \\ & {}^{\mathllap{i_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R L c &&\underset{R \tilde f}{\to}&& R d \\ \\ L c &&\underset{\tilde f}{\to}&& d }

for a unique f˜:L(c)d\tilde f:L(c)\to d – the adjunct of ff.

In particular, we have a bijection

Hom C(c,R(d))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 dd. Again, in this case there is a unique way to make LL into a functor so that this isomorphism is natural in cc as well.

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

To derive this characterization starting with a natural hom-isomorphism Hom C(L(),)Hom D(,R())Hom_{C}(L(-),-) \stackrel{\simeq}{\to} Hom_D(-,R(-)) let f˜:Lcd\tilde f : L c \to d be the image of f:cRdf : c \to R d under the bijection Hom C(c,Rd)Hom D(Lc,d)Hom_C(c, R d) \stackrel{\simeq}{\to} Hom_D(L c, d) and consider the naturality square

c,Lc Hom D(Lc,Lc) Hom C(c,RLc) Id, f˜ R(f˜)() c,d Hom D(Lc,d) Hom C(c,Rd). \array{ c , L c &&&& Hom_D(L c, L c) &\stackrel{\simeq}{\to}& Hom_C(c, R L c) \\ \uparrow^{\mathrlap{Id}}, \downarrow^\mathrlap{\tilde f} &&&& \downarrow && \downarrow^{\mathrlap{R (\tilde f)\circ(-) }} \\ c, d &&&& Hom_D(L c, d) &\stackrel{\simeq}{\to}& Hom_C(c, R d) } \,.

Let also the unit i c:cRLci_c : c \to R L c be the image of the identity Id LcId_{L c} under the hom-isomorphism and chase this identity through the commuting diagram to obtain

(LcId LcLc) (ci xRLc) (Lcf˜d) (f:ci cRLcRf˜Rd). \array{ (L c \stackrel{Id_{L c}}{\to} L c) &\mapsto& (c \stackrel{i_x}{\to} R L c) \\ \downarrow && \downarrow \\ (L c \stackrel{\tilde f}{\to} d) &\mapsto& (f : c \stackrel{i_c}{\to} R L c \stackrel{R \tilde f}{\to} R d) } \,.

Example This definition in terms of universal factorizations through the unit and counit is of particular interest in the case that RR is a full and faithful functor exhibiting DD as a reflective subcategory of CC. In this case we may think of LL as a localization and of objects in the essental image of RR 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

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

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. \,.

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 equal. 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.

For (,1)(\infty,1)-functors

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

(motivated from the discussion of correspondences in section 2.3.1)

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

Definition

Let CC and DD be quasi-categories. An adjunction between CC and DD is

For more on this see

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

General

Proposition

If a functor RR has a left adjoint LL, then LL is unique up to unique isomorphism.

If a functor LL has a right adjoint RR, then RR is unique up to unique isomorphism.

Proposition

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

Proof

Let y:IDy : I \to 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

C(x,Rlim iy i) D(Lx,lim iy i) lim iD(Lx,y i) lim iD(x,Ry i) D(x,lim iRy i), \begin{aligned} C(x, R {\lim_\leftarrow}_i y_i) & \simeq D(L x, {\lim_\leftarrow}_i y_i) \\ & \simeq {\lim_\leftarrow}_i D(L x, y_i) \\ & \simeq {\lim_\leftarrow}_i D( x, R y_i) \\ & \simeq D( x, {\lim_\leftarrow}_i R y_i) \,, \end{aligned}

where we used the adjunction isomorphism 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.

Note

A partial converse to this fact is provided by the adjoint functor theorem. See also PointwiseExpression below.

Proposition

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

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

(1)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

Let R:DCR : D \to C be a right adjoint functor such that

Then the value of the left adjoint L:CDL : C \to D on any object cc may be computed by a limit:

Lclim cRdd L c \simeq \lim_{c\to R d} d

over the comma category c/Rc/R (whose objects are pairs (d,f:cRd)(d,f:c\to R d) and whose morphisms are arrows ddd\to d' in DD making the obvious triangle commute in CC) of the projection functor

Lc=lim (c/RD). L c = \lim_{\leftarrow} (c/R \to D ) \,.

Because with this there is for every dd an obvious morphism

LRd=lim RdRddd L R d \stackrel{=}{\to} \lim_{R d \to R d'} d' \to d

(the component map over dd of the limiting cone) while moreover because RR preserves limits, we have an isomorphism

RLclim cRdRd R L c \simeq \lim_{c\to R d} R d

and hence an obvious morphism of cone tips

cRLc. c \to R L c \,.

It is easy to check that these would be the unit and counit of an adjunction LRL\dashv R.

See 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 (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

For standard references see any of the standard references on category theory, listed there, for instance

A video of a pedagogical introduction to adjoint functors is provided by

Some facts on adjoint functors are at the very beginning of

Revised on March 15, 2014 16:40:04 by Anonymous Coward (182.249.245.40)