Subsection 11.9.1 (017L): Transformations

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.9: Natural Transformations
Subsection 11.9.1: Transformations (cite)

11.9.1 Transformations

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

Definition 11.9.1.1.1 ▶ Transformations

A transformation¹ $\smash {\alpha \colon F\Rightarrow G}$ from $F$ to $G$ is a collection

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

of morphisms of $\mathcal{D}$.

¹Further Terminology: Also called an unnatural transformation for emphasis.

Notation 11.9.1.1.2 ▶ The Set of Transformations Between Two Functors

We write $\operatorname {\mathrm{Trans}}\webleft (F,G\webright )$ for the set of transformations from $F$ to $G$.

Remark 11.9.1.1.3 ▶ The Set of Transformations as a Product

We have an isomorphism

\[ \operatorname {\mathrm{Trans}}\webleft (F,G\webright )\cong \prod _{A\in \mathcal{C}}\operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F_{A},G_{A}\webright ). \]

Proof of Remark 11.9.1.1.3.

Clear.

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