Subsection 2.2.3 (0020): Binary Coproducts

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2.2.3 Binary Coproducts

Let $A$ and $B$ be sets.

Definition 2.2.3.1.1 ▶ Coproducts of Sets

The coproduct of $A$ and $B$¹ is the coproduct of $A$ and $B$ in $\mathsf{Sets}$ as in , .

¹Further Terminology: Also called the disjoint union of $A$ and $B$.

Construction 2.2.3.1.2 ▶ Construction of Coproducts of Sets

The coproduct of $A$ and $B$ is the pair $\webleft (A\coprod B,\webleft\{ \mathrm{inj}_{1},\mathrm{inj}_{2}\webright\} \webright )$ consisting of:

The Colimit. The set $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ defined by
\begin{align*} A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{z\in \webleft\{ A,B\webright\} }z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (0,a\webright )\in S\ \middle |\ a\in A\webright\} \cup \webleft\{ \webleft (1,b\webright )\in S\ \middle |\ b\in B\webright\} , \end{align*}

where $S=\webleft\{ 0,1\webright\} \times \webleft (A\cup B\webright )$.
The Cocone. The maps
\begin{align*} \mathrm{inj}_{1} & \colon A \to A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B,\\ \mathrm{inj}_{2} & \colon B \to A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B, \end{align*}

given by

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

for each $a\in A$ and each $b\in B$.

Proof of Construction 2.2.3.1.2.

We claim that $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ is the categorical coproduct of $A$ and $B$ in $\mathsf{Sets}$. Indeed, suppose we have a diagram of the form

in $\mathsf{Sets}$. Then there exists a unique map $\phi \colon A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\to C$ making the diagram

commute, being uniquely determined by the conditions

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

via

\[ \phi \webleft (x\webright )=\begin{cases} \iota _{A}\webleft (a\webright ) & \text{if $x=\webleft (0,a\webright )$,}\\ \iota _{B}\webleft (b\webright ) & \text{if $x=\webleft (1,b\webright )$} \end{cases} \]

for each $x\in A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$.

Proposition 2.2.3.1.3 ▶ Properties of Coproducts of Sets

Let $A$, $B$, $C$, and $X$ be sets.

Functoriality. The assignment $A,B,\webleft (A,B\webright )\mapsto A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ defines functors
\[ \begin{array}{ccc} A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ -_{1}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-_{2}\colon \mkern -15mu & \mathsf{Sets}\times \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}, \end{array} \]

where $-_{1}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-_{2}$ is the functor where
- Action on Objects. For each $\webleft (A,B\webright )\in \text{Obj}\webleft (\mathsf{Sets}\times \mathsf{Sets}\webright )$, we have
  
  \[ \webleft [-_{1}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-_{2}\webright ]\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B. \]
- Action on Morphisms. For each $\webleft (A,B\webright ),\webleft (X,Y\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
  
  \[ \mathbin {\textstyle \coprod _{\webleft (A,B\webright ),\webleft (X,Y\webright )}} \colon \mathsf{Sets}\webleft (A,X\webright )\times \mathsf{Sets}\webleft (B,Y\webright )\to \mathsf{Sets}\webleft (A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B,X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright ) \]
  
  of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$ at $\webleft (\webleft (A,B\webright ),\webleft (X,Y\webright )\webright )$ is defined by sending $\webleft (f,g\webright )$ to the function
  
  \[ f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g\colon A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B\to X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y \]
  
  defined by
  
  \[ \webleft [f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g\webright ]\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \webleft (0,f\webleft (a\webright )\webright ) & \text{if $x=\webleft (0,a\webright )$,}\\ \webleft (1,g\webleft (b\webright )\webright ) & \text{if $x=\webleft (1,b\webright )$,}\end{cases} \]
  
  for each $x\in A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$.
and where $A\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-$ and $-\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}B$ are the partial functors of $-_{1}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}-_{2}$ at $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Associativity. We have an isomorphism of sets
\[ \alpha ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y,Z} \colon \webleft (X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\webleft (Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Z\webright ), \]

natural in $X,Y,Z\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Unitality. We have isomorphisms of sets
\begin{align*} \lambda ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X} & \colon \text{Ø}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}X \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X,\\ \rho ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X} & \colon X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\text{Ø}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X, \end{align*}

natural in $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Commutativity. We have an isomorphism of sets
\[ \sigma ^{\mathsf{Sets},\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}_{X,Y} \colon X\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}Y \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }Y\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}X, \]

natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Symmetric Monoidality. The 7-tuple $\webleft(\phantom{\mathrlap {\alpha ^{\mathsf{Sets}}}}\mathsf{Sets}\right.$, $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$, $\text{Ø}$, $\alpha ^{\mathsf{Sets}}_{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, $\lambda ^{\mathsf{Sets}}_{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, $\rho ^{\mathsf{Sets}}_{\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}}$, $\left.\sigma ^{\mathsf{Sets}}\webright)$ is a symmetric monoidal category.