Definition 4.3.2.1.1 (00B2): Coproducts of Families of Pointed Sets

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The coproduct of the family $\smash {\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}}$, also called their wedge sum, is the pair consisting of:

The Colimit. The pointed set $\webleft (\bigvee _{i\in I}X_{i},p_{0}\webright )$ consisting of:
- The Underlying Set. The set $\bigvee _{i\in I}X_{i}$ defined by
  
  \[ \bigvee _{i\in I}X_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\coprod _{i\in I}X_{i}\webright )/\mathord {\sim }, \]
  
  where $\mathord {\sim }$ is the equivalence relation on $\coprod _{i\in I}X_{i}$ given by declaring
  
  \[ \webleft (i,x^{i}_{0}\webright )\sim \webleft (j,x^{j}_{0}\webright ) \]
  
  for each $i,j\in I$.
- The Basepoint. The element $p_{0}$ of $\bigvee _{i\in I}X_{i}$ defined by
  
  \begin{align*} p_{0} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (i,x^{i}_{0}\webright )\webright ]\\ & = \webleft [\webleft (j,x^{j}_{0}\webright )\webright ] \end{align*}
  
  for any $i,j\in I$.
The Cocone. The collection

\[ \webleft\{ \mathrm{inj}_{i} \colon \webleft (X_{i},x^{i}_{0}\webright )\to \webleft (\bigvee _{i\in I}X_{i},p_{0}\webright )\webright\} _{i\in I} \]

of morphism of pointed sets given by

\[ \mathrm{inj}_{i}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (i,x\webright ) \]

for each $x\in X_{i}$ and each $i\in I$.

We claim that $\smash {\webleft (\bigvee _{i\in I}X_{i},p_{0}\webright )}$ is the categorical coproduct of $\smash {\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}}$ 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 morphism of pointed sets

\[ \phi \colon \webleft (\bigvee _{i\in I}X_{i},p_{0}\webright )\to \webleft (C,*\webright ) \]

making the diagram

commute, being uniquely determined by the condition $\phi \circ \mathrm{inj}_{i}=\iota _{i}$ for each $i\in I$ via

\[ \phi \webleft (\webleft [\webleft (i,x\webright )\webright ]\webright )=\iota _{i}\webleft (x\webright ) \]

for each $\webleft [\webleft (i,x\webright )\webright ]\in \bigvee _{i\in I}X_{i}$, where we note that $\phi $ is indeed a morphism of pointed sets, as we have

\begin{align*} \phi \webleft (p_{0}\webright ) & = \iota _{i}\webleft (\webleft [\webleft (i,x^{i}_{0}\webright )\webright ]\webright )\\ & = *, \end{align*}

as $\iota _{i}$ is a morphism of pointed sets.