Three Approaches to Adjoints

13 Feb 2025

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The content of this text is based on Tom Leinster, Basic Category Theory.

Let $\mathcal{A}, \mathcal{B}$ be categories and $F: \mathcal{A} \to \mathcal{B}, G: \mathcal{B} \to \mathcal{A}$ be functors.

1. Definition of Adjoint Using the Naturality Axiom

$F \dashv G$ means that for any $A \in \mathcal{A}, B \in \mathcal{B}$, there exists a bijection $\Psi_{A, B} : \mathrm{Hom}_\mathcal{A}(A, G(B)) \to \mathrm{Hom}_\mathcal{B}(F(A), B)$ such that for any $p: A’ \to A, q: B \to B’$, the following commutative diagram is satisfied.

For convenience, we denote $\Psi_{A, B}(f) = \bar{f}$ for $f: A \to G(B)$, and $\Psi_{A, B}^{-1}(g) = \bar{g}$ for $g: F(A) \to B$. Accordingly, the above commutative diagram can be expressed as follows:

\[\begin{gather} \overline{A \xrightarrow{f} G(B) \xrightarrow{G(q)} G(B') } = F(A) \xrightarrow{\bar{f}} B \xrightarrow{q} B' \\ \overline{F(A') \xrightarrow{F(p)} F(A) \xrightarrow{g} B} = A' \xrightarrow{p} A \xrightarrow{\bar{g}} G(B) \\\\ \mathrm{i.e.}\\\\ \overline{G(q) \circ f} = q \circ \bar{f}\\ \overline{g \circ F(p)} = \bar{g} \circ p \end{gather}\]

The above two conditions are called the naturality axioms. From the naturality axioms, we can prove the following:

Theorem 1. The transformation $\eta: 1_{\mathcal{A}} \to GF$ defined by $\eta_A := \overline{1_{F(A)}} : A \to GF(A)$ is a natural transformation. Moreover, the transformation $\epsilon: FG \to 1_{\mathcal{B}}$ defined by $\epsilon_B := \overline{1_{G(B)}} : FG(B) \to B$ is also a natural transformation. We call $\eta$ the unit and $\epsilon$ the counit.

Proof. This follows from the following commutative diagram.

Theorem 2. For $f: A \to G(B), g: F(A) \to B$, the following holds:

\[\begin{gather} \bar{f} = \epsilon_B \circ F(f) \\ \bar{g} = G(g) \circ \eta_A \end{gather}\]

Proof. This follows from the following commutative diagram.

2. Definition of Adjoint Using Unit and Counit

$F \dashv G$ means that there exist natural transformations $\eta: 1_{\mathcal{A}} \to GF$ and $\epsilon: FG \to 1_{\mathcal{B}}$ such that for any $A \in \mathcal{A}, B \in \mathcal{B}$, the following commutative diagrams are always satisfied:

The above two conditions are called the triangle identities.

3. Definition of Adjoint Using Initial Objects

Definition. When $P: \mathcal{A} \to \mathcal{C}$ and $Q: \mathcal{B} \to \mathcal{C}$ are functors, we define the comma category $(P \Rightarrow Q)$ as follows:

• Objects: For a morphism $h: P(A) \to Q(B)$ in $\mathcal{C}$, the triplet $(A, h, B)$
• Morphisms: When the following commutative diagram holds, $(f, g): (A, h, B) \to (A’, h’, B’)$

$F \dashv G$ means that there exists a natural transformation $\eta: 1_{\mathcal{A}} \to GF$ such that for each $A \in \mathcal{A}$, when we regard $A$ as a functor $A: 1 \mapsto A$ from the singleton category $\mathbf{1}$ to $\mathcal{A}$, $\eta_A : A \to GF(A)$ is an initial object in the comma category $(A \Rightarrow G)$.

4. Proof of Equivalence

The definitions 1, 2, and 3 are all equivalent. Specifically, the following theorem holds:

Theorem 3. Let $\mathcal{A}, \mathcal{B}$ be categories and $F: \mathcal{A} \to \mathcal{B}, G: \mathcal{B} \to \mathcal{A}$ be functors. Then 1, 2, and 3 are in one-to-one correspondence.

1. A bijection $\Psi$ satisfying the naturality axioms
2. A pair of natural transformations $(\eta: 1_{\mathcal{A}} \to GF, \epsilon: FG \to 1_{\mathcal{B}})$ satisfying the triangle identities
3. A natural transformation $\eta: 1_{\mathcal{A}} \to GF$ such that for each $A \in \mathcal{A}$, $\eta_A : A \to GF(A)$ is an initial object in $(A \Rightarrow G)$

Proof. We show that 1 and 2 are in one-to-one correspondence, and that 2 and 3 are in one-to-one correspondence.

1 and 2 are in one-to-one correspondence.

Given $\Psi$, we define $\eta$ and $\epsilon$ as the unit and counit respectively. That $\eta$ and $\epsilon$ satisfy the triangle identities follows easily from Theorem 2.

Conversely, suppose we are given a pair of natural transformations $(\eta, \epsilon)$ satisfying the triangle identities. From this, we define $\Psi$ as follows. For $f: A \to G(B), g: F(A) \to B$:

\[\begin{gather} \Psi_{A, B}(f) = \bar{f} = \epsilon_B \circ F(f) \\ \Psi_{A, B}^{-1} = \bar{g} = G(g) \circ \eta_A \end{gather}\]

First, we show that $\Psi_{A, B}^{-1}$ is indeed the inverse of $\Psi_{A, B}$, i.e., that $\bar{\bar{g}} = g, \bar{\bar{f}} = f$. The former follows from the following commutative diagram. The latter can be obtained by replacing $F, G$ with $G, F$ respectively in the diagram and then taking the appropriate dual.

Now we show that $\Psi$ satisfies the naturality axioms. We shall prove only one axiom. For $f: A \to G(B), q: B \to B’$, we show that $\overline{G(q) \circ f} = q \circ \bar{f}$. First,

\[\overline{G(q) \circ f} = \epsilon_{B'} \circ FG(q) \circ F(f)\]

and

\[q \circ \bar{f} = q \circ \epsilon_B \circ F(f)\]

Therefore, it suffices to show that $\epsilon_{B’} \circ FG(q) = q \circ \epsilon_B$. This is nothing other than the definition of a natural transformation. Hence $\Psi$ satisfies the naturality axioms.

2 and 3 are in one-to-one correspondence.

Given a pair of natural transformations $(\eta, \epsilon)$ satisfying the triangle identities, let us show that for each $A \in \mathcal{A}$, $\eta_A : A \to GF(A)$ is an initial object in $(A \Rightarrow G)$.

Conversely, when $\eta$ is a natural transformation such that for each $A \in \mathcal{A}$, $\eta_A : A \to GF(A)$ is an initial object in $(A \Rightarrow G)$, let us show that there exists a unique $\epsilon: FG \to 1_{\mathcal{B}}$ such that $(\eta, \epsilon)$ satisfies the triangle identities.