Subsection 4.2.3 (00A9): Products — The Clowder Project

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

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The product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pair consisting of:

The Limit. The pointed set $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$.
The Cone. The morphisms of pointed sets

\begin{align*} \text{pr}_{1} & \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to \webleft (X,x_{0}\webright ),\\ \text{pr}_{2} & \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to \webleft (Y,y_{0}\webright ) \end{align*}

defined by

\begin{align*} \text{pr}_{1}\webleft (x,y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x,\\ \text{pr}_{2}\webleft (x,y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y \end{align*}

for each $\webleft (x,y\webright )\in X\times Y$.

We claim that $\webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )$ is the categorical product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ in $\mathsf{Sets}_{*}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets

\[ \phi \colon \webleft (P,*\webright )\to \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \]

making the diagram

commute, being uniquely determined by the conditions

\begin{align*} \text{pr}_{1}\circ \phi & = p_{1},\\ \text{pr}_{2}\circ \phi & = p_{2} \end{align*}

via

\[ \phi \webleft (x\webright )=\webleft (p_{1}\webleft (x\webright ),p_{2}\webleft (x\webright )\webright ) \]

for each $x\in P$. Note that this is indeed a morphism of pointed sets, as we have

\begin{align*} \phi \webleft (*\webright ) & = \webleft (p_{1}\webleft (*\webright ),p_{2}\webleft (*\webright )\webright )\\ & = \webleft (x_{0},y_{0}\webright ),\end{align*}

where we have used that $p_{1}$ and $p_{2}$ are morphisms of pointed sets.

Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, and $\webleft (Z,z_{0}\webright )$ be pointed sets.

Functoriality. The assignments
\[ \webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )\mapsto \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \]

define functors

\[ \begin{array}{ccc} A\times -\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\times B\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\times -_{2}\colon \mkern -15mu & \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}, \end{array} \]

defined in the same way as the functors of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.1.3.1.3.
Lack of Adjointness. The functors $X\times -$ and $-\times Y$ do not admit right adjoints.
Associativity. We have an isomorphism of pointed sets
\[ \webleft (\webleft (X\times Y\webright )\times Z,\webleft (\webleft (x_{0},y_{0}\webright ),z_{0}\webright )\webright ) \cong \webleft (X\times \webleft (Y\times Z\webright ),\webleft (x_{0},\webleft (y_{0},z_{0}\webright )\webright )\webright ) \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Unitality. We have isomorphisms of pointed sets
\begin{align*} \webleft (\text{pt},\star \webright )\times \webleft (X,x_{0}\webright ) & \cong \webleft (X,x_{0}\webright ),\\ \webleft (X,x_{0}\webright )\times \webleft (\text{pt},\star \webright ) & \cong \webleft (X,x_{0}\webright ), \end{align*}

natural in $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Commutativity. We have an isomorphism of pointed sets
\[ \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright ) \cong \webleft (Y\times X,\webleft (y_{0},x_{0}\webright )\webright ), \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\times ,\webleft (\text{pt},\star \webright )\webright )$ is a symmetric monoidal category.

4.2.3 Products