Subsection 2.3.5 (003A): Sets of Maps

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.3: Operations With Sets
Subsection 2.3.5: Sets of Maps (cite)

2.3.5 Sets of Maps

Let $A$ and $B$ be sets.

Definition 2.3.5.1.1 ▶ Sets of Maps

The set of maps from $A$ to $B$^[1] is the set $\textup{Hom}_{\mathsf{Sets}}\webleft (A,B\webright )$^[2] whose elements are the functions from $A$ to $B$.

Proposition 2.3.5.1.2 ▶ Properties of Sets of Maps

Let $A$ and $B$ be sets.

Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \textup{Hom}_{\mathsf{Sets}}\webleft (X,Y\webright )$ define functors
\begin{gather*} \begin{aligned} \textup{Hom}_{\mathsf{Sets}}\webleft (X,-\webright ) & \colon \mathsf{Sets}\to \mathsf{Sets},\\ \textup{Hom}_{\mathsf{Sets}}\webleft (-,Y\webright ) & \colon \mathsf{Sets}^{\mathsf{op}} \to \mathsf{Sets}, \end{aligned}\\ \textup{Hom}_{\mathsf{Sets}}\webleft (-_{1},-_{2}\webright ) \colon \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}\to \mathsf{Sets}. \end{gather*}

Proof of Proposition 2.3.5.1.2.

Item 1: Functoriality

This follows from Chapter 8: Categories, Item 2 and Item 5 of Proposition 8.1.6.1.2.