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.
We discuss here the definition of adjointness of functors $L \dashv R$ in terms of a natural bijection between hom-sets (Def. below):
We show that this is equivalent to the abstract definition, in terms of an adjunction in the 2-category Cat, in Prop. below.
$\,$
(adjoint functors in terms of natural bijections of hom-sets)
Let $\mathcal{C}$ and $\mathcal{D}$ be two categories, and let
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:
This means that for all objects $c \in \mathcal{C}$ and $d \in \mathcal{D}$ there is a bijection of hom-sets
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
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
(adjunction unit and counit in terms of hom-isomorphism)
Given a pair of adjoint functors
according to Def. one says that
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
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
(general adjuncts in terms of unit/counit)
Consider a pair of adjoint functors
according to Def. , with adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ according to Def. .
Then
The adjunct $\widetilde f$ of any morphism $L(c) \overset{f}{\to} d$ is obtained from $R$ and $\eta_c$ as the composite
Conversely, the adjunct $f$ of any morphism $c \overset{\widetilde f}{\longrightarrow} R(d)$ is obtained from $L$ and $\epsilon_d$ as
The adjunction units $\eta_c$ and adjunction counits $\epsilon_d$ are components of natural transformations of the form
and
The adjunction unit and adjunction counit satisfy the triangle identities, saying that
and
For the first statement, consider the naturality square (2) in the form
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:
For the second statement, we have to show that for every morphism $f \colon c_1 \to c_2$ the following square commutes:
To see this, consider the naturality square (2) in the form
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.
(adjointness in terms of hom-isomorphism equivalent to adjunction in $Cat$)
Two functors
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
there exist natural transformations
and
which satisfy the triangle identities
and
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
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:
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:
adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. below;
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.
(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
hence for each object $c \in \mathcal{C}$ a bijection
such that for each morphism $g \;\colon\; c_2 \to c_1$, the following diagram commutes
(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.
Notice that
in the language of presheaves the assumption is that for each $d \in \mathcal{D}$ the presheaf
is represented by the object $R(d)$, and naturally so.
In terms of the Yoneda embedding
we have
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
This manifestly has a unique solution
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.
In more fancy language, the statement of Prop. is the following:
By precomposition $L$ defines a functor of presheaf categories
By restriction along the Yoneda embedding $y \;\colon\; \mathcal{D} \to [\mathcal{D}^{op}, Set]$ this yields the functor
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
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
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.
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.
$\,$
(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
an object $L(c)\in \mathcal{D}$
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
for a unique $\widetilde f \;\colon\; L(c) \longrightarrow d$, to be called the adjunct of $f$.
(e.g Borceux, vol 1, Def. 3.1.1)
(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:
$c \overset{\eta_c}{\to} R(L(c))$ is a universal morphism into $R(L(c))$ (Def. );
$(c, \eta_c)$ is the initial object in the comma category $c/R$.
(collection of universal arrows equivalent to adjoint functor)
Let $R \;\colon\; \mathcal{D} \to \mathcal{C}$ be a functor. Then the following are equivalent:
$R$ has a left adjoint functor $L \colon \mathcal{C} \to \mathcal{D}$ according to Def. ,
for every object $c \in \mathcal{C}$ there is a universal arrow $c \overset{\eta_c}{\longrightarrow} R(L(c))$, according to Def. .
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
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
(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
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 profunctor
defines a category $C *^k D$ with $Obj(C *^k D) = Obj(C) \sqcup Obj(D)$ and with hom set given by
($k(X,Y)$ is also called the heteromorphisms).
This category naturally comes with a functor to the interval category
Now, every functor $L : C \to D$ induces a profunctor
and every functor $R : D \to C$ induces a profunctor
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.
Functors $L : C \to D$ and $R : D \to C$ are adjoint precisely if we have a commutative diagram
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.
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
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:
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
or, in terms of left Kan liftings:
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.
(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.
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
As in the proof of Prop. , the Yoneda lemma implies that
for some isomorphism
But then the uniqueness statement of Prop. implies that the collection of these isomorphisms for each object constitues a natural isomorphism between the functors.
(left adjoints preserve colimits and right adjoints preserve limits)
Let $(L \dashv R) \colon \mathcal{D} \to \mathcal{C}$ be a pair of adjoint functors (Def. ). Then
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 $x \in C$
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
The argument that shows the preservation of colimits by $L$ is analogous.
A partial converse to Prop. is provided by the adjoint functor theorem. See also Pointwise Expression below.
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$\phantom{A}$ |
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
being an injection for all objects $a$. Then use that, by adjointness, we have an isomorphism
and that, by the formula for adjuncts and the zig-zag identity, this is such that the composite
is the component map of the functor $R$:
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
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
is an injection. Analogously to the previous argument we find that this is equivalent to
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
is a surjection, for all objects $a$.
The proof of the other statements proceeds analogously.
Parts of this statement can be strenghened:
Let $(L \dashv R) : D \to C$ be a pair of adjoint functors such that there is any natural isomorphism
then also the counit $\epsilon : L R \to Id$ is an isomorphism.
This appears as (Johnstone, lemma 1.1.1).
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 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
for each object $d \in \mathcal{D}$, the limit of the canonical functor out of the comma category of $R$ under $d$
exists.
In this case the value of the left adjoint $L$ on $d$ is given by that limit:
(e.g. MacLane, chapter X, theorem 2)
First assume that the left adjoint exist. Then
$R$ is a right adjoint and hence preserves limits since all right adjoints preserve limits;
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
Since the $d \overset{f}{\to} R(d)$ constitute a cone over the diagram of the $R(d)$, there is universal morphism
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
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. 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.
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.
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 $p$
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
between $C$ and the category of presheaves on $K$, where
the nerve functor is given by
and the realization functor is given by the coend
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.
adjoint functor, adjunction
For the basics, see any text on category theory (and see the references at adjunction), for instance:
Francis Borceux, Vol 1, Section 3 of Handbook of Categorical Algebra
Peter Johnstone, first pages of Sketches of an Elephant
Though the definition of an adjoint equivalence appears in Grothendieck's Tohoku paper, the idea of adjoint functors in general goes back to
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)
See also
Wikipedia, Adjoint Functors
“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 January 3, 2021 at 10:36:41. See the history of this page for a list of all contributions to it.