Construction 2.1.2.1.2 (01DE): Construction of the Product of a Family of Sets

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Style 4	`book`	Computer Modern	`amsthm`

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 $x\in P$.