Item 1a (0111) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.5: Functors
Subsection 11.5.4: The Natural Transformation Associated to a Functor
Item 1a (cite)

11.5.4 The Natural Transformation Associated to a Functor

Definition 11.5.4.1.1 ▶ The Natural Transformation Associated to a Functor

Every functor $F\colon \mathcal{C}\to \mathcal{D}$ defines a natural transformation¹

called the natural transformation associated to $F$, consisting of the collection

\[ \left\{ F^{\dagger }_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \right\} _{\webleft (A,B\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}^{\mathsf{op}}\times \mathcal{C}\webright )} \]

with

\[ F^{\dagger }_{A,B} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}F_{A,B}. \]

¹This is the $1$-categorical version of Chapter 4: Constructions With Sets, of .

Proof of Definition 11.5.4.1.1.

The naturality condition for $F^{\dagger }$ is the requirement that for each morphism

\[ \webleft (\phi ,\psi \webright ) \colon \webleft (X,Y\webright ) \to \webleft (A,B\webright ) \]

of $\mathcal{C}^{\mathsf{op}}\times \mathcal{C}$, the diagram

acting on elements as

commutes, which follows from the functoriality of $F$.

Proposition 11.5.4.1.2 ▶ Properties of Natural Transformations Associated to Functors

Let $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ be functors.

1.
Interaction With Natural Isomorphisms. The following conditions are equivalent:

(a)
The natural transformation $F^{\dagger }\colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\Longrightarrow {\operatorname {\mathrm{Hom}}_{\mathcal{D}}}\circ {\webleft (F^{\mathsf{op}}\times F\webright )}$ associated to $F$ is a natural isomorphism.

(b)

The functor $F$ is fully faithful.

Interaction With Composition. We have an equality of pasting diagrams

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

\[ \webleft (G\circ F\webright )^{\dagger }=\webleft (G^{\dagger }\mathbin {\star }\operatorname {\mathrm{id}}_{F^{\mathsf{op}}\times F}\webright )\circ F^{\dagger }. \]

Interaction With Identities. We have

\[ \operatorname {\mathrm{id}}^{\dagger }_{\mathcal{C}}=\operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )}, \]

i.e. the natural transformation associated to $\operatorname {\mathrm{id}}_{\mathcal{C}}$ is the identity natural transformation of the functor $\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )$.

Proof of Proposition 11.5.4.1.2.

Item 1: Interaction With Natural Isomorphisms

Clear.

Item 2: Interaction With Composition

Clear.

Item 3: Interaction With Identities

Clear.