category theory

# Contents

## Definition

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

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

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

### In terms of Hom isomorphism

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

$Hom_D(L(-),-) \simeq Hom_C(-,R(-)) \,.$

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

$Hom_D(L(c),d) \simeq Hom_C(c,R(d))$

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

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

### In terms of representable functors

#### Global definition

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

$Hom_D(L(-),d) \cong Hom_C(-,R(d)).$

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

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

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

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

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

such that

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

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

$\bar L \simeq Y \circ R \,.$

#### Local definition

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

it may happen that

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

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

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

$\array{ && c \\ & {}^{\mathllap{i_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R L c &&\underset{R \tilde f}{\to}&& R d \\ \\ L c &&\underset{\tilde f}{\to}&& d }$

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

In particular, we have a bijection

$Hom_C(c,R(d)) \cong Hom_D(L(c),d)$

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

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

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

$\array{ c , L c &&&& Hom_D(L c, L c) &\stackrel{\simeq}{\to}& Hom_C(c, R L c) \\ \uparrow^{\mathrlap{Id}}, \downarrow^\mathrlap{\tilde f} &&&& \downarrow && \downarrow^{\mathrlap{R (\tilde f)\circ(-) }} \\ c, d &&&& Hom_D(L c, d) &\stackrel{\simeq}{\to}& Hom_C(c, R d) } \,.$

Let also the unit $i_c : c \to R L c$ be the image of the identity $Id_{L c}$ under the hom-isomorphism and chase this identity through the commuting diagram to obtain

$\array{ (L c \stackrel{Id_{L c}}{\to} L c) &\mapsto& (c \stackrel{i_x}{\to} R L c) \\ \downarrow && \downarrow \\ (L c \stackrel{\tilde f}{\to} d) &\mapsto& (f : c \stackrel{i_c}{\to} R L c \stackrel{R \tilde f}{\to} R d) } \,.$

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

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

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

This category naturally comes with a functor to the interval category

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

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 equal. 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.

### For $(\infty,1)$-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 $C$ and $D$ be quasi-categories. An adjunction between $C$ and $D$ is

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

• which is

• such that $C \simeq K_{0}$ and $D\simeq K_{1}$.

For more on this see

### 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

### General

###### Proposition

If a functor $R$ has a left adjoint $L$, then $L$ is unique up to unique isomorphism.

If a functor $L$ has a right adjoint $R$, then $R$ is unique up to unique isomorphism.

###### Proposition

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

• $L$ preserves all colimits that may exist in $C$,

• $R$ preserves all limits in $D$.

###### Proof

Let $y : I \to D$ 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} 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 $x$ 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.

###### Note

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

###### Proposition

Let $L \dashv R$ be a pair of adjoint functors. 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 an 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.

###### 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 $R$:

\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

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

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 strenghened:

###### 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

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

• $C$ is a locally small category,

• $D$ has all small limits.

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

$L c \simeq \lim_{c\to R d} d$

over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor

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

Because with this there is for every $d$ an obvious morphism

$L R d \stackrel{=}{\to} \lim_{R d \to R d'} d' \to d$

(the component map over $d$ of the limiting cone) while moreover because $R$ preserves limits, we have an isomorphism

$R L c \simeq \lim_{c\to R d} R d$

and hence an obvious morphism of cone tips

$c \to R L c \,.$

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

See 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.

## 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 $(L \dashv R)$ 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.

## 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)