9.2 Unit and counit of an adjunction

Adjunctions in category theory connect two categories through functors. The unit and counit are key natural transformations that define this relationship. They map objects between categories, showing how they relate.

Understanding unit and counit is crucial for grasping adjunctions. These transformations satisfy specific conditions, ensuring the adjunction works properly. They help us see how objects in different categories correspond to each other.

Unit and Counit of an Adjunction

Definition of unit and counit

• Given an adjunction $F \dashv G$ between categories $\mathcal{C}$ and $\mathcal{D}$, the unit is a natural transformation $\eta: 1_\mathcal{C} \to GF$ and the counit is a natural transformation $\varepsilon: FG \to 1_\mathcal{D}$
• The unit has a component $\eta_X: X \to GF(X)$ for each object $X$ in $\mathcal{C}$ ($\mathcal{C}$ could be the category of sets and $X$ a specific set)
• The counit has a component $\varepsilon_Y: FG(Y) \to Y$ for each object $Y$ in $\mathcal{D}$ ($\mathcal{D}$ could be the category of vector spaces and $Y$ a specific vector space)

Role in adjunctions

• The unit $\eta$ and counit $\varepsilon$ relate objects in $\mathcal{C}$ and $\mathcal{D}$ via the adjoint functors $F$ and $G$, establishing a connection between the categories
• The unit $\eta$ embeds an object $X$ in $\mathcal{C}$ into the image of $\mathcal{D}$ under $G$, i.e., $GF(X)$, acting as a "canonical map" from $X$ to $GF(X)$ (e.g., embedding a set into its free vector space)
• The counit $\varepsilon$ projects an object $FG(Y)$ in the image of $\mathcal{C}$ under $F$ back to the original object $Y$ in $\mathcal{D}$, acting as a "canonical map" from $FG(Y)$ to $Y$ (e.g., projecting a vector space onto its underlying set)
• The unit and counit satisfy coherence conditions (triangle identities) ensuring compatibility with the adjunction

Triangle identities proof

• Given an adjunction $F \dashv G$ with unit $\eta$ and counit $\varepsilon$, the triangle identities are:
  1. For each object $X$ in $\mathcal{C}$: $G(\varepsilon_{F(X)}) \circ \eta_{G(F(X))} = 1_{G(F(X))}$
  2. For each object $Y$ in $\mathcal{D}$: $\varepsilon_{F(G(Y))} \circ F(\eta_{G(Y)}) = 1_{F(G(Y))}$
• Proof of the first identity:
  1. Start with the adjunction isomorphism: $\text{Hom}_\mathcal{D}(F(X), Y) \cong \text{Hom}_\mathcal{C}(X, G(Y))$
  2. Set $Y = F(X)$ and apply the isomorphism to $1_{F(X)}$ to obtain $\eta_X$
  3. Apply $G$ to both sides of $\varepsilon_{F(X)} \circ F(\eta_X) = 1_{F(X)}$ to obtain the first identity
• The second identity is proven similarly by setting $X = G(Y)$ and applying the adjunction isomorphism to $1_{G(Y)}$

Determination of adjunction isomorphism

• Given functors $F: \mathcal{C} \to \mathcal{D}$ and $G: \mathcal{D} \to \mathcal{C}$ with natural transformations $\eta: 1_\mathcal{C} \to GF$ and $\varepsilon: FG \to 1_\mathcal{D}$ satisfying the triangle identities, the adjunction isomorphism can be reconstructed:
  1. For objects $X$ in $\mathcal{C}$ and $Y$ in $\mathcal{D}$, define $\varphi: \text{Hom}_\mathcal{D}(F(X), Y) \to \text{Hom}_\mathcal{C}(X, G(Y))$ by $\varphi(f) = G(f) \circ \eta_X$
  2. Define $\psi: \text{Hom}_\mathcal{C}(X, G(Y)) \to \text{Hom}_\mathcal{D}(F(X), Y)$ by $\psi(g) = \varepsilon_Y \circ F(g)$
• To show $\varphi$ and $\psi$ are inverses:
  1. For $f: F(X) \to Y$, $\psi(\varphi(f)) = \varepsilon_Y \circ F(G(f) \circ \eta_X) = (\varepsilon_Y \circ FG(f)) \circ F(\eta_X) = f \circ (\varepsilon_{F(X)} \circ F(\eta_X)) = f \circ 1_{F(X)} = f$
  2. For $g: X \to G(Y)$, $\varphi(\psi(g)) = G(\varepsilon_Y \circ F(g)) \circ \eta_X = (G(\varepsilon_Y) \circ GF(g)) \circ \eta_X = G(\varepsilon_Y) \circ (GF(g) \circ \eta_X) = G(\varepsilon_Y) \circ \eta_{G(Y)} \circ g = 1_{G(Y)} \circ g = g$
• Thus, the unit and counit determine the adjunction up to isomorphism (e.g., the free-forgetful adjunction between sets and groups)