Adjunctions - Category Theory - Cambridge III notes

3 Adjunctions

Definition 3.1 (Adjunction, D. Kan 1958). Let $C$ and $D$ be categories. An adjunction between $C$ and $D$ consists of functors $F : C \to D$ and $G : D \to C$ together with, for each $A \in ob C$ and $B \in ob D$ , a bijection between morphisms $F A \to B$ in $D$ and morphisms $A \to G B$ in $C$ , which is natural in $A$ and $B$ . (If $C$ and $D$ are locally small, this means that $D (F ∙, ∙)$ and $C (∙, G ∙)$ are naturally isomorphic functors $C^{o p} \times D \to S e t$ .)

We say $F$ is left adjoint to $G$ , or $G$ is right adjoint to $F$ , and we write $(F ⊣ G)$ .

Example 3.2.

(a) The free functor $F : S e t \to G p$ is left adjoint to the forgetful functor $G p \overset{U}{\to} S e t$ . By definition, homomorphisms $F A \to G$ correspond to functions $A \to U G$ ; naturality in $A$ was built into the definition of $F$ in Example 1.5(b) and naturality in $G$ is immediate.
(b) The forgetful functor $U : T o p \to S e t$ has a left adjoint $D$ , which equips a set $A$ with its discrete topology since any function $A \to U X$ is continuous as a map $D A \to X$ . $U$ also has a right adjoint $I$ given by the ‘ìndiscrete’ topology.
(c) The functor $ob : C a t \to S e t$ has a left adjoint $D$ given by discrete categories, and a right adjoint $I$ : $I A$ is the category with objects $A$ and morphisms $a \to b$ for each $(a, b)$ . $D$ also has a left adjoint $π_{0}$ : $π_{0} C$ is the set of connected components of $C$ , i.e. the quotient of $ob C$ by the smallest equivalence relation which identifies $dom f$ with $cod f$ for all $f \in mor C$ .
(d) Given a set $A$ , we can regard $(∙) \times A$ as a functor $S e t \to S e t$ . It has a right adjoint, namely $S e t (A, ∙)$ . Given $f : B \times A \to C$ we can regard it as a function $λ f : B \to S e t (A, C)$ by $λ f (b) (a) = f (b, a)$ .
We call a category $C$ cartesian closed if it has binary products as defined in Example 2.6(f) and each $(∙) \times A$ has a right adjoint ${(∙)}^{A}$ . For example, $C a t$ is cartesian clsosed, with $D^{C}$ taken to be the $[C, D]$ .
(e) Let $M = {1, e}$ be the $2$ -element monoid with $e^{2} = e$ (and identity $1$ ). We have a functor $F : S e t \to [M, S e t]$ sending $A$ to $(A, 1_{A})$ and a functor $G : [M, S e t] \to S e t$ sending $(A, e)$ to ${a \in A | e a = a}$ .
We have $(F ⊣ G ⊣ F)$ : $(F ⊣ G)$ since any $f : M \to (B, e)$ takes values in $G (B, e)$ and any $g : (B, e) \to F A$ is determined by its restriction to $G (B, e)$ since $g (b) = g (e, b)$ . However, note that this is not an equivalence of categories.
(f) Let $1$ be the category with one object and one morphism (which must the identity on the only object). A left adjoint for the unique functor $C \to 1$ picks out an initial object of $C$ , i.e. an object such that there is a unique $I \to A$ for each $A \in ob C$ . Dually, a right adjoint for $C \to 1$ ‘is’ a terminal object of $C$ (a terminal object is an initial object in $C^{o p}$ ).
Again, the example of $G p$ shows that these two can coincide.
(g) Suppose given $A \overset{f}{\to} B$ in $S e t$ . We have order-preserving mappings $P f : P A \to P B$ and $P^{*} f : P B \to P A$ , and $(P f ⊣ P^{*} f$ since $A^{'} \subseteq f^{- 1} B^{'} ⟺ f (A^{'}) \subseteq B^{'}$ .
(h) Suppose given a relation $R \subseteq A \times B$ . We define ${(∙)}^{r} : P A \to P B$ and ${(∙)}^{l} : P B \to P A$ by $\begin{array}{l} {(S)}^{r} & = {b \in B | (\forall a \in S) ((a, b) \in R)} \\ {(T)}^{l} & = {a \in A | (\forall b \in T) ((a, b) \in R)} \end{array}$
These are contravariant functors and $S \subseteq T^{l} ⟺ S \times T \subseteq R ⟺ T \subseteq S^{r}$ . We say ${(∙)}^{r}$ and ${(∙)}^{l}$ are adjoint on the right.
(i) $P^{*} : {S e t}^{o p} \to S e t$ is self-adjoint on the right, since functions $A \to P B$ and functions $B \to P A$ both correspond to relations $R \subseteq A \times B$ .
(j) ${(∙)}^{*} : {V e c t}_{k}^{*} \to {V e c t}_{k}$ is self-adjoint on the right, since linear maps $V \to W^{*}$ and $W \to V^{*}$ both correspond to bilinear maps $V \times W \to k$ .

Theorem 3.3. Assuming that:

$G : D \to C$ is a functor
for $A \in ob C$ , let $(A ↓ G)$ be the category whose objects are pairs $(B, f)$ where $B \in ob D$ and $f : A \to G B$ , and whose morphisms $(B, f) \to (B^{'}, f^{'})$ are morphisms $g : B \to B^{'}$ making
commute.

Then specifying a left adjoint for

G

is equivalent to specifying an initial object of

(A ↓ G)

for each

A

Proof. First suppose $(F ⊣ G)$ . For each $A \in ob C$ , let $η_{A} : A \to G F A$ be the morphism corresponding to $1_{F A} : F A \to F A$ . Then $(F A, η_{A})$ is an initial object of $(A ↓ G)$ : given any $f : A \to G B$ , the diagram

commutes if and only if

g

corresponds to

f

under the adjunction, by naturality of the adjunction bijection.

So there’s a unique morphism $(F A, η_{A}) \to (B, f)$ in $(A ↓ G)$ .

Conversely, suppose given in initial object $(F A, η_{A})$ in $(A ↓ G)$ for each $A$ . We make $F$ into a function $C \to D$ : given $A \overset{f}{\to} B$ , $F f$ is the unique morphism $(F A, η_{A}) \to (F B, η_{B} f)$ in $(A ↓ G)$ . Functoriality comes from uniqueness: given $B \overset{g}{\to} C$ , $(F g) (F f)$ and $F (g f)$ are both morphisms $(F A, η_{A}) \to (F C, η_{C} g f)$ in $(A ↓ G)$ . The adjunction bijection sends $A \overset{f}{\to} G B$ to the unique morphism $(F A, η_{A}) \to (B, f)$ in $(A ↓ G)$ , with inverse sending $F A \overset{g}{\to} B$ to $(G g) η_{A} : A \to G B$ . This is natural in $A$ since $η$ is a natural transformation $1_{C} \to G F$ and natural in $B$ since $G$ is functorial. □

Corollary 3.4. Assuming that:

$F$ and $F^{'}$ are both left adjoint to $G : D \to C$

Then there is a canonical natural isomorphism

α : F \to F^{'}

Proof. $(F A, η_{A})$ and $(F^{'} A, η_{A}^{'})$ are both initial in $(A ↓ G)$ , so there’s a unique isomorphism $α_{A}$ between them. $α$ is natural: given $A \overset{f}{\to} B$ , $(F^{'} f) α_{A}$ and $α_{B} (F f)$ are both morphisms $(F A, η_{A}) \to (F^{'} B, η_{B}^{'} f)$ in $(A ↓ G)$ , so they’re equal. □

As a result of this, we will often talk about “the” left adjoint of a functor (when it exists), because we don’t usually care about which one in the isomorphism class we use.

Lemma 3.5. Assuming that:

$C ⇄_{G}^{F} D ⇄_{K}^{H} E$
$(F ⊣ G)$ and $(H ⊣ K)$

Then

(H F ⊣ G K)

Proof. Given $A \in ob C$ , $C \in ob E$ , we have bijections between morphisms $H F A \to C$ , morphisms $F A \to K C$ , and morphisms $A \to G K C$ which are both natural in $A$ and $C$ , $D$ . □

Corollary 3.6. Assuming that:

the diagram
is a commutative square of categories and functors
all the functors have left adjoints

Then the square of left adjoints commutes up to natural isomorphism.

Proof. By Lemma 3.5, both ways round are left adjoint to $H F = K G$ , so by Corollary 3.4 they’re isomorphic. □

We saw in Theorem 3.3 that an adjoint $(F ⊣ G)$ gives rise to a natural transformation $η : 1_{C} \to G F$ , called the unit of the adjunction. Dually, we have $𝜀 : F G \to 1_{D}$ , the counit of $(F ⊣ G)$ .

Theorem 3.7. Assuming that:

$F : C \to D$ and $G : D \to C$ are functors

Then specifying an adjunction

(F ⊣ G)

is equivalent to specifying a natural transformation

η : 1_{C} \to G F

and

𝜀 : F G \to 1_{D}

satisfying the two commutative diagrams:

Proof. Suppose $(F ⊣ G)$ . We defined $η$ in the proof of Theorem 3.3, and $𝜀$ is defined dually. Since $𝜀_{F A}$ corresponds to $1_{G F A}$ , the composite $𝜀_{F A} (F η_{A})$ corresponds to $1_{G F A} η_{A} = η_{A}$ . But by definition $1_{F A}$ corresponds to $η_{A}$ . The other identity is dual.

Conversely, suppose given $η$ and $𝜀$ satisfying the triangular identities. Given $F A \overset{f}{\to} B$ , we define $Φ (f) = (G f) η_{A} : A \to G F A \to G B$ . Dually, given $A \overset{g}{\to} G B$ , we define $Ψ (g) = 𝜀_{B} (F g)$ . Then $Ψ Φ (f) = Ψ ((G f) η_{A}) = 𝜀_{B} (F G f) F η_{A} = f (𝜀_{F A}) (F η_{A}) = f$ , and dually $Φ Ψ (g) = g$ . Naturality of $Φ$ and $Ψ$ follows from naturality of $η$ and $𝜀$ . □

In Definition 1.9, we had natural isomorphisms $α : 1_{C} \to G F$ and $β : F G \to 1_{D}$ . These look like the unit and counit of an adjunction $(F ⊣ G)$ : do they satisfy the triangular identities? No, but we can always change them:

Proposition 3.8. Assuming that:

$F : C \to D$ , $G : D \to C$ , $α : 1_{C} \to G F$ and $β : F G \to 1_{D}$ be an equivalence of categories as defined in Definition 1.9

Then there exist isomorphisms

α^{'} : 1_{C} \to G F

and

β^{'} : F G \to 1_{D}

satisfying the triangular identities. In particular,

(F ⊣ G ⊣ F)

Proof. We define $α^{'} = α$ and take $β^{'}$ to be the composite

F G \overset{{(F G β)}^{- 1}}{\to} F G F G \overset{{(F α_{G})}^{- 1}}{\to} F G \overset{β}{\to} 1_{D} .

Note that $F G β = β_{F G}$ , since

commutes by naturality of

β

, and

β

is monic. Similarly,

G F α = α_{G F}

To verify the triangular identities, consider

F FGF FGF GF

F FGF

−1−1−−11 −1
F1(F((1βαFβFαFβFFFGα)FG)F) =(FGFα) F

which commutes by naturality of

β^{- 1}

For the second triangular identity, we have

G GF G GF GF G

G GF G

−1−1 −−11 −1
α1(α((1GGGGGGGGβFFβGα)βG)) = (αGFG) G β

Hence by Theorem 3.7 we have

(F ⊣ G)

. But

{(β^{'})}^{- 1}

and

α^{- 1}

also satisfy the triangular identities for and adjunction

(G ⊣ F)

. □

Lemma 3.9. Assuming that:

$(F : C \to D ⊣ G : D \to C)$ an adjunction with counit $𝜀$

Then

(i) $G$ is faithful if and only if $𝜀$ is pointwise epic
(ii) $G$ is full and faithful if and only if $𝜀$ is an isomorphism

Proof.

(1) Given $g : B \to C$ in $D$ , $g 𝜀_{B}$ corresponds to $G g$ under the adjunction. So $𝜀_{B}$ epic if and only if $G$ acts injectively on morphisms with domain $B$ and specified codomain. Hence $𝜀_{B}$ epic for all $B$ if and only if $G$ is faithful.
(2) Similarly, $G$ full and faithful if and only if for all $B$ and $C$ composition with $𝜀_{B}$ is a bijection $D (B, C) \to D (F G B, C)$ . This happens if and only if $𝜀_{b} : F G B \to B$ is an isomorphism for all $B$ . □

Definition 3.10 (Reflection). By a reflection, we mean an adjunction such that the right adjoint is full and faithful (equivalently: the counit is an isomorphism). We say $D \subseteq C$ is a reflective subcategory if it’s full and the inclusion $D \to C$ has a left adjoint.

Example 3.11.

(a) $A b G p$ is reflective in $G p$ : the left adjoint to the inclusion sends $G$ to $G ∕ G^{'}$ where $G^{'}$ is the subgroup generated by commutators. Any homomorphism $G \to A$ with $A$ abelian factors uniquely through the quotient map $G \to G ∕ G^{'}$ .
(b) Recall that a group $G$ is torsion if all elements have finite order, and torsion free if its only element of finite order is $1$ . In an abelian group $A$ , the torsion leements form a subgroup $A_{t}$ , and $A \mapsto A_{t}$ is right adjoint to the inclusion $t A b G p \to A b G p$ , since any homomorphism $B \to A$ whose $B$ is torsion takes values in $A_{t}$ . Similarly, $A \mapsto A ∕ A_{t}$ defines a left adjoint to the inclusion $t f A b G p \to A b G p$ .
(c) Let $K H a u s \subseteq T o p$ be the full subcategory of compact Hausdorff spaces. $K H a u s$ is reflective in $T o p$ : the left adjoint is the Stone-Čech compactification $β$ .
(d) Let $S e q \subseteq T o p$ be the full subcategory of sequential spaces, i.e. those in which all sequentially closed sets are closed. The inclusion $S e q \to T o p$ has a right adjoint sending $X$ to $X_{s}$ , the same set as $X$ with all sequentially closed sets declared to be closed. The identity mapping $X_{s} \to X$ is (continuous, and) the counit of the adjunction.
(e) The category $P r e o r d$ of preordered sets is reflective in $C a t$ : the reflection sends $C$ to $C ∕ ≃$ where $≃$ is the congruence identifying all paralell pairs in $C$ .
(f) Given a topological space $X$ , the poset $Ω (X)$ of open subsets of $X$ is coreflective in $P (X)$ , since if $U$ is open and $A \subseteq X$ is arbitrary, we have $U \subseteq A$ if and only if $U \subseteq A^{\circ}$ (recall $^{\circ}$ denotes interior). Dually, the poset of closed subsets is reflective in $P (X)$ .