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.

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.

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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:

for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_2: S_2 \to X$:

there exists a unique mapping $h: C \to X$ such that:

$h \circ i_1 = f_1$

$h \circ i_2 = f_2$

Hence:

$\begin{xy} \xymatrix@L+2mu@+1em{ & C \ar@{-->}[dd]_*{h} & \\ S_1 \ar[ru]^*{i_1} \ar[rd]_*{f_1} & & S_2 \ar[lu]_*{i_2} \ar[ld]^*{f_2} \\ & X & }\end{xy}$

is a commutative diagram.

Also see

• 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 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Definition $3.3$