8.8.4 Horizontal Composition of Natural Transformations

Definition 8.8.4.1.1 ▶ Horizontal Composition of Natural Transformations

The horizontal composition[1][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

Proof of Definition 8.8.4.1.1.

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.[3]

Definition 8.8.4.1.2 ▶ 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 transformation[4]
    \[ \text{id}_{G}\star \alpha \colon G\circ \phi \Longrightarrow G\circ \psi . \]
  2. The right whiskering of $\alpha $ with $F$ is the natural transformation[5]
    \[ \alpha \star \text{id}_{F}\colon \phi \circ F\Longrightarrow \psi \circ F. \]

Proposition 8.8.4.1.3 ▶ 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 8.8.4.1.3.
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.

Footnotes

[1] Further Terminology: Also called the Godement product of $\alpha $ and $\beta $.
[2] Horizontal composition forms a map
\[ \mathbin {\star }_{\webleft (F,H\webright ),\webleft (G,K\webright )}\colon \text{Nat}\webleft (H,K\webright )\times \text{Nat}\webleft (F,G\webright )\to \text{Nat}\webleft (H\circ F,K\circ G\webright ). \]
[3] Reference: Proposition 1.3.4 of [Borceux, Handbook of Categorical Algebra I].
[4] Further Notation: Also written $G\alpha $ or $G\mathbin {\star }\alpha $, although we won’t use either of these notations in this work.
[5] Further Notation: Also written $\alpha F$ or $\alpha \mathbin {\star }F$, although we won’t use either of these notations in this work.
Previous
Next

View Subsection 8.8.4 as PDF

Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: