# Definition:Coproduct

*Not to be confused with Definition:Comultiplication.*

## Definition

Let $\mathbf C$ be a metacategory.

Let $A$ and $B$ be objects of $\mathbf C$.

A **coproduct diagram** for $A$ and $B$ comprises an object $P$ and morphisms $i_1: A \to P$, $i_2: B \to P$:

- $\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{P} = "P", <4em,0em>*+{B} = "B", "A";"P" **@{-} ?>*@{>} ?*!/^.8em/{i_1}, "B";"P" **@{-} ?>*@{>} ?*!/_.8em/{i_2}, \end{xy}$

subjected to the following universal mapping property:

- $\begin{xy} <-4em,0em>*+{A} = "A", <0em,0em>*+{X} = "X", <4em,0em>*+{B} = "B", "A";"X" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, \end{xy}$

- there is a unique morphism $u: P \to X$ such that:

- $\begin{xy} <0em,5em>*+{X} = "X", <-5em,0em>*+{A} = "A", <0em,0em>*+{P} = "P", <5em,0em>*+{B} = "B", "A";"X" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "B";"X" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, "P";"X" **@{--} ?>*@{>} ?*!/_.6em/{u}, "A";"P" **@{-} ?>*@{>} ?*!/_.8em/{i_1}, "B";"P" **@{-} ?>*@{>} ?*!/^.8em/{i_2}, \end{xy}$

- is a commutative diagram.

That is:

- $x_1 = u \circ i_i$ and $x_2 = u \circ i_2$

In this situation, $P$ is called a **coproduct of $A$ and $B$** and may be denoted $A + B$.

We generally write $\sqbrk {x_1, x_2}$ for the unique morphism $u$ determined by above diagram.

The morphisms $i_1$ and $i_2$ are often taken to be implicit.

They are called **injections**; if necessary, $i_1$ can be called the **first injection** and $i_2$ the **second injection**.

### Coproduct of Sets

When the objects $A$ and $B$ are sets, the definition of **coproduct** takes on the following form.

Let $S_1$ and $S_2$ be sets.

A **coproduct** $\struct {C, i_1, i_2}$ of $S_1$ and $S_2$ comprises a set $C$ together with mappings $i_1: S_1 \to C$, $i_2: S_2 \to C$ such that:

Hence:

- $\begin{xy} \[email protected][email protected]+1em{ & C \[email protected]{-->}[dd]_*{h} & \\ S_1 \ar[ru]^*{i_1} \ar[rd]_*{f_1} & & S_2 \ar[lu]_*{i_2} \ar[ld]^*{f_2} \\ & X & }\end{xy}$

- is a commutative diagram.

## Also see

### Examples

- Coproduct of Free Monoids
- Disjoint Union is Coproduct in Category of Sets
- Coproduct of Ordered Sets
- Supremum is Coproduct in Order Category

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Definition $3.3$