Coproduct - Knowlpedia

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

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

\iota_1:A\to A\sqcup B,\qquad \iota_2:B\to A\sqcup B

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

[f,g]:A\sqcup B\to X

making

[f,g]\circ \iota_1=f,\qquad [f,g]\circ \iota_2=g

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

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

$\mathbf{Grp}$ . 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 $H\to X$ .

$\mathbf{Ab}$ 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.