Definition 6.3.3.1.1 (00B6): Coproducts of Pointed Sets

6.3.3 Coproducts

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

Definition 6.3.3.1.1 ▶ Coproducts of Pointed Sets

The coproduct of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$¹ is the coproduct of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ in $\mathsf{Sets}_{*}$ as in , .

¹Further Terminology: Also called the wedge sum of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$.

Construction 6.3.3.1.2 ▶ Construction of Coproducts of Pointed Sets

Concretely, 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 $y\in Y$.

Proof of Construction 6.3.3.1.2.

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.

Proposition 6.3.3.1.3 ▶ Properties of Wedge Sums of Pointed Sets

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

1.
Functoriality. The assignments

\[ \webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )\mapsto \webleft (X\vee Y,p_{0}\webright ) \]

define functors

\begin{align*} X\vee - & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -\vee Y & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -_{1}\vee -_{2} & \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{align*}
2.
Associativity. We have an isomorphism of pointed sets

\[ \webleft (X\vee Y\webright )\vee Z\cong X\vee \webleft (Y\vee Z\webright ), \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \mathsf{Sets}_{*}$.
3.
Unitality. We have isomorphisms of pointed sets

\begin{align*} \webleft (\mathrm{pt},*\webright )\vee \webleft (X,x_{0}\webright ) & \cong \webleft (X,x_{0}\webright ),\\ \webleft (X,x_{0}\webright )\vee \webleft (\mathrm{pt},*\webright ) & \cong \webleft (X,x_{0}\webright ),\end{align*}

natural in $\webleft (X,x_{0}\webright )\in \mathsf{Sets}_{*}$.
4.
Commutativity. We have an isomorphism of pointed sets

\[ X\vee Y \cong Y\vee X, \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \mathsf{Sets}_{*}$.
5.
Symmetric Monoidality. The triple $\webleft (\mathsf{Sets}_{*},\vee ,\mathrm{pt}\webright )$ is a symmetric monoidal category.
6.
The Fold Map. We have a natural transformation
called the fold map, whose component

\[ \nabla _{X} \colon X\vee X \to X \]

at $X$ is given by

\[ \nabla _{X}\webleft (p\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} x & \text{if $p=\webleft [\webleft (0,x\webright )\webright ]$,}\\ x & \text{if $p=\webleft [\webleft (1,x\webright )\webright ]$} \end{cases} \]

for each $p\in X\vee X$.

Proof of Proposition 6.3.3.1.3.

Item 1: Functoriality

This follows from

Item 2: Associativity

Omitted.

Item 3: Unitality

Omitted.

Item 4: Commutativity

Omitted.

Item 5: Symmetric Monoidality

Omitted.

Item 6: The Fold Map

Naturality for the transformation $\nabla $ is the statement that, given a morphism of pointed sets $f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$, we have

Indeed, we have

\begin{align*} \webleft [\nabla _{Y}\circ \webleft (f\vee f\webright )\webright ]\webleft (\webleft [\webleft (i,x\webright )\webright ]\webright ) & = \nabla _{Y}\webleft (\webleft [\webleft (i,f\webleft (x\webright )\webright )\webright ]\webright )\\ & = f\webleft (x\webright )\\ & = f\webleft (\nabla _{X}\webleft (\webleft [\webleft (i,x\webright )\webright ]\webright )\webright )\\ & = \webleft [f\circ \nabla _{X}\webright ]\webleft (\webleft [\webleft (i,x\webright )\webright ]\webright ) \end{align*}

for each $\webleft [\webleft (i,x\webright )\webright ]\in X\vee X$, and thus $\nabla $ is indeed a natural transformation.

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