Adjunctions - Category Theory - Cambridge III notes

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 $V \times W \to k$ .

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. □

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.

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)

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:

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)

. □

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 $P (X)$ .

3 Adjunctions