9.9.5 Horizontal Composition of Natural Transformations

Definition 9.9.5.1.1 ▶ Horizontal Composition of Natural Transformations

The horizontal composition1,2 of two natural transformations $\alpha \colon F\Longrightarrow G$ and $\beta \colon H\Longrightarrow K$ as in the diagram

of $\alpha $ and $\beta $ is the natural transformation

\[ \beta \mathbin {\star }\alpha \colon \webleft (H\circ F\webright )\Longrightarrow \webleft (K\circ G\webright ), \]

as in the diagram

consisting of the collection

\[ \webleft\{ \webleft (\beta \mathbin {\star }\alpha \webright )_{A} \colon H\webleft (F\webleft (A\webright )\webright ) \to K\webleft (G\webleft (A\webright )\webright ) \webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )}, \]

of morphisms of $\mathcal{E}$ with

1Further Terminology: Also called the Godement product of $\alpha $ and $\beta $.
2Horizontal composition forms a map

\[ \mathbin {\star }_{\webleft (F,H\webright ),\webleft (G,K\webright )}\colon \text{Nat}\webleft (H,K\webright )\times \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (H\circ F,K\circ G\webright ). \]

Proof of Definition 9.9.5.1.1.

First, we claim that we indeed have

This is, however, simply the naturality square for $\beta $ applied to the morphism $\alpha _{A}\colon F\webleft (A\webright )\to G\webleft (A\webright )$. Next, we check the naturality condition for $\beta \mathbin {\star }\alpha $, which is the requirement that the boundary of the diagram

commutes. Since

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

so does the boundary diagram. Hence $\beta \circ \alpha $ is a natural transformation.1

1Reference: Proposition 1.3.4 of [Borceux, Handbook of Categorical Algebra I].

Definition 9.9.5.1.2 ▶ Whiskering of Functors With Natural Transformations

Let

be a diagram in $\mathsf{Cats}_{\mathsf{2}}$.

  1. The left whiskering of $\alpha $ with $G$ is the natural transformation1
    \[ \text{id}_{G}\star \alpha \colon G\circ \phi \Longrightarrow G\circ \psi . \]
  2. The right whiskering of $\alpha $ with $F$ is the natural transformation2
    \[ \alpha \star \text{id}_{F}\colon \phi \circ F\Longrightarrow \psi \circ F. \]

1Further Notation: Also written $G\alpha $ or $G\mathbin {\star }\alpha $, although we won’t use either of these notations in this work.
2Further Notation: Also written $\alpha F$ or $\alpha \mathbin {\star }F$, although we won’t use either of these notations in this work.

Proposition 9.9.5.1.3 ▶ Properties of Horizontal Composition of Natural Transformations

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

  1. Functionality. The assignment $\webleft (\beta ,\alpha \webright )\mapsto \beta \mathbin {\star }\alpha $ defines a function
    \[ \mathbin {\star }_{\webleft (F,G\webright ),\webleft (H,K\webright )}\colon \text{Nat}\webleft (H,K\webright )\times \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (H\circ F,K\circ G\webright ). \]
  2. Associativity. Let

    be a diagram in $\mathsf{Cats}_{\mathsf{2}}$. The diagram

    commutes, i.e. given natural transformations

    we have

    \[ \webleft (\gamma \mathbin {\star }\beta \webright )\mathbin {\star }\alpha =\gamma \mathbin {\star }\webleft (\beta \mathbin {\star }\alpha \webright ). \]
  3. Interaction With Identities. Let $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ be functors. The diagram

    commutes, i.e. we have

    \[ \text{id}_{G}\mathbin {\star }\text{id}_{F}=\text{id}_{G\circ F}. \]
  4. 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 9.9.5.1.3.
Item 1: Functionality
Clear.
Item 2: Associativity
Omitted.
Item 3: Interaction With Identities
We have
\begin{align*} \webleft (\text{id}_{G}\mathbin {\star }\text{id}_{F}\webright )_{A} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\text{id}_{G}\webright )_{F_{A}}\circ G_{\webleft (\text{id}_{F}\webright )_{A}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{G_{F_{A}}}\circ G_{\text{id}_{F_{A}}}\\ & = \text{id}_{G_{F_{A}}}\circ \text{id}_{G_{F_{A}}}\\ & = \text{id}_{G_{F_{A}}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\text{id}_{G\circ F}\webright )_{A} \end{align*}

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

Item 4: Middle Four Exchange
Let $A\in \text{Obj}\webleft (\mathcal{C}\webright )$ and consider the diagram
The top composition
is given by $\webleft (\webleft (\beta '\circ \beta \webright )\mathbin {\star }\webleft (\alpha '\circ \alpha \webright )\webright )_{A}$, while the bottom composition
is given by $\webleft (\webleft (\beta '\mathbin {\star }\alpha '\webright )\circ \webleft (\beta \mathbin {\star }\alpha \webright )\webright )_{A}$. Now, Subdiagram (1) corresponds to the naturality condition
for $\beta \colon G_{1}\Longrightarrow G_{2}$ at $\alpha '_{A}\colon F_{2}\webleft (A\webright )\to F_{3}\webleft (A\webright )$, and thus commutes. Thus we have
\[ \webleft (\webleft (\beta '\circ \beta \webright )\mathbin {\star }\webleft (\alpha '\circ \alpha \webright )\webright )_{A} = \webleft (\webleft (\beta '\mathbin {\star }\alpha '\webright )\circ \webleft (\beta \mathbin {\star }\alpha \webright )\webright )_{A} \]

for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$ and therefore

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

This finishes the proof.

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