Subsection 4.2.2 (001W): Coproducts of Families of Sets

The coproduct of $\left\{ A_{i}\right\} _{i\in I}$¹ is the coproduct of $\left\{ A_{i}\right\} _{i\in I}$ in $\mathsf{Sets}$ as in , .

¹Further Terminology: Also called the disjoint union of the family $\left\{ A_{i}\right\} _{i\in I}$.

Concretely, the disjoint union of $\left\{ A_{i}\right\} _{i\in I}$ is the pair $(\coprod _{i\in I}A_{i},\left\{ \mathrm{inj}_{i}\right\} _{i\in I})$ consisting of:

1.
The Colimit. The set $\coprod _{i\in I}A_{i}$ defined by

\[ \coprod _{i\in I}A_{i}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ (i,x)\in I\times \left(\bigcup _{i\in I}A_{i}\right)\ \middle |\ \text{$x\in A_{i}$}\right\} . \]
2.
The Cocone. The collection

\[ \left\{ \mathrm{inj}_{i} \colon A_{i}\to \coprod _{i\in I}A_{i}\right\} _{i\in I} \]

of maps given by

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

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

We claim that $\coprod _{i\in I}A_{i}$ is the categorical coproduct of $\left\{ A_{i}\right\} _{i\in I}$ 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 map $\phi \colon \coprod _{i\in I}A_{i}\to C$ making the diagram

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

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

for each $(i,x)\in \coprod _{i\in I}A_{i}$.

Let $\left\{ A_{i}\right\} _{i\in I}$ be a family of sets.

1.
Functoriality. The assignment $\left\{ A_{i}\right\} _{i\in I}\mapsto \coprod _{i\in I}A_{i}$ defines a functor

\[ \coprod _{i\in I}\colon \mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets})\to \mathsf{Sets} \]

where
- •
  Action on Objects. For each $(A_{i})_{i\in I}\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets}))$, we have
  
  \[ \left[\coprod _{i\in I}\right]((A_{i})_{i\in I})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\coprod _{i\in I}A_{i} \]
- •
  Action on Morphisms. For each $(A_{i})_{i\in I},(B_{i})_{i\in I}\in \operatorname {\mathrm{Obj}}(\mathsf{Fun}(I_{\mathsf{disc}},\mathsf{Sets}))$, the action on $\operatorname {\mathrm{Hom}}$-sets
  
  \[ \left(\coprod _{i\in I}\right)_{(A_{i})_{i\in I},(B_{i})_{i\in I}} \colon \operatorname {\mathrm{Nat}}((A_{i})_{i\in I},(B_{i})_{i\in I})\to \mathsf{Sets}\left(\coprod _{i\in I}A_{i},\coprod _{i\in I}B_{i}\right) \]
  
  of $\coprod _{i\in I}$ at $((A_{i})_{i\in I},(B_{i})_{i\in I})$ is defined by sending a map
  
  \[ \left\{ f_{i}\colon A_{i}\to B_{i} \right\} _{i\in I} \]
  
  in $\operatorname {\mathrm{Nat}}((A_{i})_{i\in I},(B_{i})_{i\in I})$ to the map of sets
  
  \[ \coprod _{i\in I}f_{i}\colon \coprod _{i\in I}A_{i}\to \coprod _{i\in I}B_{i} \]
  
  defined by
  
  \[ \left[\coprod _{i\in I}f_{i}\right](i,a) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{i}(a) \]
  
  for each $(i,a)\in \coprod _{i\in I}A_{i}$.

Item 1: Functoriality

This follows from

The Clowder Project

4.2.2 Coproducts of Families of Sets