Item 1 (017Y) — The Clowder Project

11.9.4 Vertical Composition of Natural Transformations

Definition 11.9.4.1.1 ▶ Vertical Composition of Natural Transformations

The vertical composition of two natural transformations $\alpha \colon F\Longrightarrow G$ and $\beta \colon G\Longrightarrow H$ as in the diagram

is the natural transformation $\beta \circ \alpha \colon F\Longrightarrow H$ consisting of the collection

\[ \left\{ \webleft (\beta \circ \alpha \webright )_{A} \colon F\webleft (A\webright ) \to H\webleft (A\webright ) \right\} _{A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )} \]

with

\[ \webleft (\beta \circ \alpha \webright )_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\beta _{A}\circ \alpha _{A} \]

for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.

Proof of Definition 11.9.4.1.1.

The naturality condition for $\beta \circ \alpha $ is the requirement that the boundary of the diagram

commutes. Since

•
Subdiagram (1) commutes by the naturality of $\alpha $.
•
Subdiagram (2) commutes by the naturality of $\beta $.

so does the boundary diagram. Hence $\beta \circ \alpha $ is a natural transformation.

Proposition 11.9.4.1.2 ▶ Properties of Vertical Composition of Natural Transformations

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

1.
Functionality. The assignment $\webleft (\beta ,\alpha \webright )\mapsto \beta \circ \alpha $ defines a function

\[ \circ _{F,G,H}\colon \operatorname {\mathrm{Nat}}\webleft (G,H\webright )\times \operatorname {\mathrm{Nat}}\webleft (F,G\webright )\to \operatorname {\mathrm{Nat}}\webleft (F,H\webright ). \]

Associativity. Let $F,G,H,K\colon \mathcal{C}\overset {\rightrightarrows }{\rightrightarrows }\mathcal{D}$ be functors. The diagram

commutes, i.e. given natural transformations

\[ F\mathbin {\overset {\alpha }{\Longrightarrow }}G\mathbin {\overset {\beta }{\Longrightarrow }}H\mathbin {\overset {\gamma }{\Longrightarrow }}K, \]

we have

\[ \webleft (\gamma \circ \beta \webright )\circ \alpha =\gamma \circ \webleft (\beta \circ \alpha \webright ). \]

Unitality. Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.

(a)
Left Unitality. The diagram
commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have

\[ \operatorname {\mathrm{id}}_{G}\circ \alpha =\alpha . \]
(b)
Right Unitality. The diagram
commutes, i.e. given a natural transformation $\alpha \colon F\Longrightarrow G$, we have

\[ \alpha \circ \operatorname {\mathrm{id}}_{F}=\alpha . \]

Middle Four Exchange. Let $F_{1},F_{2},F_{3}\colon \mathcal{C}\to \mathcal{D}$ and $G_{1},G_{2},G_{3}\colon \mathcal{D}\to \mathcal{E}$ be functors. The diagram

commutes, i.e. given a diagram

in $\mathsf{Cats}_{\mathsf{2}}$, we have

\[ \webleft (\beta '\mathbin {\star }\alpha '\webright )\circ \webleft (\beta \mathbin {\star }\alpha \webright )=\webleft (\beta '\circ \beta \webright )\mathbin {\star }\webleft (\alpha '\circ \alpha \webright ). \]

Proof of Proposition 11.9.4.1.2.

Item 1: Functionality

Clear.

Item 2: Associativity

Indeed, we have

\begin{align*} \webleft (\webleft (\gamma \circ \beta \webright )\circ \alpha \webright )_{A} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\gamma \circ \beta \webright )_{A}\circ \alpha _{A}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\gamma _{A}\circ \beta _{A}\webright )\circ \alpha _{A}\\ & = \gamma _{A}\circ \webleft (\beta _{A}\circ \alpha _{A}\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\gamma _{A}\circ \webleft (\beta \circ \alpha \webright )_{A}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\gamma \circ \webleft (\beta \circ \alpha \webright )\webright )_{A} \end{align*}

for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, showing the desired equality.

Item 3: Unitality

We have

\begin{align*} \webleft (\operatorname {\mathrm{id}}_{G}\circ \alpha \webright )_{A} & = \operatorname {\mathrm{id}}_{G}\circ \alpha _{A}\\ & = \alpha _{A},\\ \webleft (\alpha \circ \operatorname {\mathrm{id}}_{F}\webright )_{A} & = \alpha _{A}\circ \operatorname {\mathrm{id}}_{F}\\ & = \alpha _{A} \end{align*}

for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, showing the desired equality.

Item 4: Middle Four Exchange

This is proved in Item 4 of Proposition 11.9.5.1.3.

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