Definition 4.3.3.1.1 (00B6): Coproducts of Pointed Sets

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The coproduct of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$, also called their wedge sum, is the pair consisting of:

The Colimit. The pointed set $\webleft (X\vee Y,p_{0}\webright )$ consisting of:
- The Underlying Set. The set $X\vee Y$ defined by
  where $\mathord {\sim }$ is the equivalence relation on $X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y$ obtained by declaring $\webleft (0,x_{0}\webright )\sim \webleft (1,y_{0}\webright )$.
- The Basepoint. The element $p_{0}$ of $X\vee Y$ defined by
  
  \begin{align*} p_{0} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (0,x_{0}\webright )\webright ]\\ & = \webleft [\webleft (1,y_{0}\webright )\webright ]. \end{align*}
The Cocone. The morphisms of pointed sets

\begin{align*} \mathrm{inj}_{1} & \colon \webleft (X,x_{0}\webright ) \to \webleft (X\vee Y,p_{0}\webright ),\\ \mathrm{inj}_{2} & \colon \webleft (Y,y_{0}\webright ) \to \webleft (X\vee Y,p_{0}\webright ), \end{align*}

given by

\begin{align*} \mathrm{inj}_{1}\webleft (x\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (0,x\webright )\webright ],\\ \mathrm{inj}_{2}\webleft (y\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (1,y\webright )\webright ], \end{align*}

for each $x\in X$ and each $y\in Y$.

We claim that $\webleft (X\vee Y,p_{0}\webright )$ is the categorical coproduct 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

\[ \phi \colon \webleft (X\vee Y,p_{0}\webright )\to \webleft (C,*\webright ) \]

making the diagram

commute, being uniquely determined by the conditions

\begin{align*} \phi \circ \mathrm{inj}_{X} & = \iota _{X},\\ \phi \circ \mathrm{inj}_{Y} & = \iota _{Y} \end{align*}

via

\[ \phi \webleft (z\webright )=\begin{cases} \iota _{X}\webleft (x\webright ) & \text{if $z=\webleft [\webleft (0,x\webright )\webright ]$ with $x\in X$,}\\ \iota _{Y}\webleft (y\webright ) & \text{if $z=\webleft [\webleft (1,y\webright )\webright ]$ with $y\in Y$} \end{cases} \]

for each $z\in X\vee Y$, where we note that $\phi $ is indeed a morphism of pointed sets, as we have

\begin{align*} \phi \webleft (p_{0}\webright ) & = \iota _{X}\webleft (\webleft [\webleft (0,x_{0}\webright )\webright ]\webright )\\ & = \iota _{Y}\webleft (\webleft [\webleft (1,y_{0}\webright )\webright ]\webright )\\ & = *, \end{align*}

as $\iota _{X}$ and $\iota _{Y}$ are morphisms of pointed sets.