Subsection 9.9.6 (018C): Properties of Natural Transformations

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Table of Contents
Part 3: Category Theory
Chapter 9: Categories
Section 9.9: Natural Transformations
Subsection 9.9.6: Properties of Natural Transformations (cite)

9.9.6 Properties of Natural Transformations

Proposition 9.9.6.1.1 ▶ Natural Transformations as Categorical Homotopies

Let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors. The following data are equivalent:¹

A natural transformation $\alpha \colon F\Longrightarrow G$.
A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\to \mathcal{D}^{\mathbb {1}}$ filling the diagram
A functor $\webleft [\alpha \webright ]\colon \mathcal{C}\times \mathbb {1}\to \mathcal{D}$ filling the diagram

¹Taken from [MO 64365].

Proof of Proposition 9.9.6.1.1.

From Item 1 to Item 2 and Back

We may identify $\mathcal{D}^{\mathbb {1}}$ with $\mathsf{Arr}(\mathcal{D})$. Given a natural transformation $\alpha \colon F\Longrightarrow G$, we have a functor

making the diagram in Item 2 commute. Conversely, every such functor gives rise to a natural transformation from $F$ to $G$, and these constructions are inverse to each other.

From Item 2 to Item 3 and Back

This follows from Item 3 of Proposition 9.10.1.1.2.

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