Coproduct

Let $\mathcal C$ be a category and let $A,B$ be objects of $\mathcal C$.

Definition

A (binary) coproduct of $A$ and $B$ is a triple $(A\sqcup B,\iota_1,\iota_2)$ consisting of an object $A\sqcup B$ and morphisms

such that for every object $X$ and every pair of morphisms $f:A\to X$, $g:B\to X$, there exists a unique morphism

making

hold (see composition ).

Coproducts are unique up to unique isomorphism .

Coproduct is the dual notion to product : a coproduct in $\mathcal C$ is a product in the opposite category $\mathcal C^{\mathrm{op}}$. It is a special case of a colimit .

Examples

1. $\mathbf{Set}$.
   The coproduct is the disjoint union $A\sqcup B$, with injections $A\hookrightarrow A\sqcup B$ and $B\hookrightarrow A\sqcup B$.

2. $\mathbf{Grp}$.
   The coproduct of groups $G,H$ is the free product $G\ast H$. A homomorphism $G\ast H\to X$ is uniquely determined by its restrictions $G\to X$ and $H\to X$.

3. $\mathbf{Ab}$ and $R$-Mod.
   The coproduct is the direct sum $A\oplus B$. In these additive settings, finite coproducts and finite products coincide (biproducts), a key feature of an additive category .