4.3.3 Coproducts

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

Definition 4.3.3.1.1 ▶ Coproducts of Pointed Sets

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 Definition 4.3.3.1.1.

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 4.3.3.1.2 ▶ 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 (\text{pt},*\webright )\vee \webleft (X,x_{0}\webright ) & \cong \webleft (X,x_{0}\webright ),\\ \webleft (X,x_{0}\webright )\vee \webleft (\text{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 $\webleft (\mathsf{Sets}_{*},\vee ,\text{pt}\webright )$ 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 4.3.3.1.2.
Item 1: Functoriality
This follows from , of .
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.

Previous
Next

View Subsection 4.3.3 as PDF

Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: