Subsection 2.1.2 (000N): Products of Families of Sets

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

The product¹ of $\webleft\{ A_{i}\webright\} _{i\in I}$ is the product of $\webleft\{ A_{i}\webright\} _{i\in I}$ in $\mathsf{Sets}$ as in , .

¹Further Terminology: Also called the Cartesian product of $\webleft\{ A_{i}\webright\} _{i\in I}$.

The product of $\webleft\{ A_{i}\webright\} _{i\in I}$ is the pair $\webleft (\prod _{i\in I}A_{i},\webleft\{ \text{pr}_{i}\webright\} _{i\in I}\webright )$ consisting of:

The Limit. The set $\prod _{i\in I}A_{i}$ defined by
\[ \prod _{i\in I}A_{i} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \mathsf{Sets}\webleft(I,\bigcup _{i\in I}A_{i}\webright)\ \middle |\ \text{for each $i\in I$, we have $f\webleft (i\webright )\in A_{i}$}\webright\} . \]
The Cone. The collection
\[ \webleft\{ \text{pr}_{i} \colon \prod _{i\in I}A_{i}\to A_{i}\webright\} _{i\in I} \]

of maps given by

\[ \text{pr}_{i}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (i\webright ) \]

for each $f\in \prod _{i\in I}A_{i}$ and each $i\in I$.

We claim that $\prod _{i\in I}A_{i}$ is the categorical product of $\webleft\{ A_{i}\webright\} _{i\in I}$ in $\mathsf{Sets}$. Indeed, suppose we have, for each $i\in I$, a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon P\to \prod _{i\in I}A_{i}$ making the diagram

commute, being uniquely determined by the condition $\text{pr}_{i}\circ \phi =p_{i}$ for each $i\in I$ via

\[ \phi \webleft (x\webright )=\webleft (p_{i}\webleft (x\webright )\webright )_{i\in I} \]

for each $x\in P$.

Less formally, we may think of Cartesian products and projection maps as follows:

We think of $\prod _{i\in I}A_{i}$ as the set whose elements are $I$-indexed collections $\webleft (a_{i}\webright )_{i\in I}$ with $a_{i}\in A_{i}$ for each $i\in I$.
We view the projection maps
\[ \webleft\{ \text{pr}_{i} \colon \prod _{i\in I}A_{i}\to A_{i}\webright\} _{i\in I} \]

as being given by

\[ \text{pr}_{i}\webleft (\webleft (a_{j}\webright )_{j\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a_{i} \]

for each $\webleft (a_{j}\webright )_{j\in I}\in \prod _{i\in I}A_{i}$ and each $i\in I$.

Let $\webleft\{ A_{i}\webright\} _{i\in I}$ be a family of sets.

Functoriality. The assignment $\webleft\{ A_{i}\webright\} _{i\in I}\mapsto \prod _{i\in I}A_{i}$ defines a functor
\[ \prod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\to \mathsf{Sets} \]

where
- Action on Objects. For each $\webleft (A_{i}\webright )_{i\in I}\in \text{Obj}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, we have
  
  \[ \webleft[\prod _{i\in I}\webright]\webleft (\webleft (A_{i}\webright )_{i\in I}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\prod _{i\in I}A_{i} \]
- Action on Morphisms. For each $\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\in \text{Obj}\webleft (\mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}\webright )\webright )$, the action on $\textup{Hom}$-sets
  
  \[ \webleft(\prod _{i\in I}\webright)_{\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}} \colon \text{Nat}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )\to \mathsf{Sets}\webleft(\prod _{i\in I}A_{i},\prod _{i\in I}B_{i}\webright) \]
  
  of $\prod _{i\in I}$ at $\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ is defined by sending a map
  
  \[ \webleft\{ f_{i}\colon A_{i}\to B_{i} \webright\} _{i\in I} \]
  
  in $\text{Nat}\webleft (\webleft (A_{i}\webright )_{i\in I},\webleft (B_{i}\webright )_{i\in I}\webright )$ to the map of sets
  
  \[ \prod _{i\in I}f_{i}\colon \prod _{i\in I}A_{i}\to \prod _{i\in I}B_{i} \]
  
  defined by
  
  \[ \webleft[\prod _{i\in I}f_{i}\webright]\webleft (\webleft (a_{i}\webright )_{i\in I}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{i}\webleft (a_{i}\webright )\webright )_{i\in I} \]
  
  for each $\webleft (a_{i}\webright )_{i\in I}\in \prod _{i\in I}A_{i}$.

Item 1: Functoriality

This follows from

The Clowder Project

2.1.2 Products of Families of Sets