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:

For any object $X$ and morphisms $x_1, x_2$ like so:

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

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. I.e., $x_1 = u \circ i_i$ 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 $\left[{x_1, x_2}\right]$ 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.

Also see

• Product, the dual notion
• Coproduct is Unique

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 ... (previous) ... (next): $\S 3.2$: Definition $3.3$