Adjunction

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

Definition. An adjunction between $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ together with a natural isomorphism

for all $c \in \mathrm{Ob}(\mathcal{C})$ and $d \in \mathrm{Ob}(\mathcal{D})$, natural in both variables. We call $F$ the left adjoint and $G$ the right adjoint, and write $F \dashv G$.

Equivalently, an adjunction $F \dashv G$ is determined by natural transformations $\eta: \mathrm{id}_{\mathcal{C}} \Rightarrow G \circ F$ (the unit) and $\varepsilon: F \circ G \Rightarrow \mathrm{id}_{\mathcal{D}}$ (the counit) satisfying the triangle identities:

Proposition. The two definitions are equivalent. Given the hom-set adjunction $\varphi$, the unit and counit are $\eta_c = \varphi_{c,F(c)}(\mathrm{id}_{F(c)})$ and $\varepsilon_d = \varphi^{-1}_{G(d),d}(\mathrm{id}_{G(d)})$. Conversely, given $(\eta, \varepsilon)$ satisfying the triangle identities, $\varphi_{c,d}(f) = G(f) \circ \eta_c$ defines the natural isomorphism.

Proposition. Every adjunction $F \dashv G$ gives rise to a monad $(T, \eta, \mu)$ on $\mathcal{C}$ where $T = G \circ F$, $\eta$ is the unit of the adjunction, and $\mu = G\varepsilon_F: GFGF \Rightarrow GF$; and dually a comonad $(K, \varepsilon, \delta)$ on $\mathcal{D}$ where $K = F \circ G$.

Proof sketch. Associativity of $\mu$ and the unit laws follow from the triangle identities and naturality of $\eta$ and $\varepsilon$.

Proposition. Left adjoints preserve all colimits; right adjoints preserve all limits.

Proof sketch. For $F \dashv G$ and a diagram $D: I \to \mathcal{D}$ with limit $\lim D$, the chain of natural isomorphisms $\mathrm{Hom}(c, G(\lim D)) \cong \mathrm{Hom}(Fc, \lim D) \cong \lim \mathrm{Hom}(Fc, D(-)) \cong \lim \mathrm{Hom}(c, GD(-)) \cong \mathrm{Hom}(c, \lim GD)$ exhibits $G(\lim D) \cong \lim GD$ by the Yoneda lemma. The colimit case is dual.

Proposition (Uniqueness). If $F \dashv G$ and $F \dashv G'$, then $G \cong G'$ by a canonical natural isomorphism. That is, adjoints are unique up to unique natural isomorphism.

Proof sketch. For each $d$, the functors $\mathrm{Hom}(F(-), d)$ are representable by both $G(d)$ and $G'(d)$, so $G(d) \cong G'(d)$ by Yoneda.

Examples.

• Free-forgetful: The free group functor $F: \mathbf{Set} \to \mathbf{Grp}$ is left adjoint to the forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$. The unit sends a set $S$ to the inclusion $S \hookrightarrow U(F(S))$.
• Tensor-hom: For a commutative ring $R$ and an $R$-module $M$, the functor $M \otimes_R (-)$ is left adjoint to $\mathrm{Hom}_R(M, -)$.
• Sheafification: The sheafification functor $a: \mathrm{PSh}(\mathcal{C}) \to \mathrm{Sh}(\mathcal{C})$ is left adjoint to the inclusion functor $i: \mathrm{Sh}(\mathcal{C}) \hookrightarrow \mathrm{PSh}(\mathcal{C})$.
• Residuation: In a Heyting algebra $H$ viewed as a thin category, the meet functor $a \wedge (-)$ is left adjoint to the Heyting implication $a \to (-)$. That is, $a \wedge b \leq c$ if and only if $b \leq (a \to c)$.