Biproduct

Definition

For two objects

Suppose $C$ is a category and $A_{1}, A_{2} \in O b C$ . A biproduct of $A_{1}$ and $A_{2}$ is an object that serves the role of both a product and a coproduct. More explicitly, it is an object $C$ along with maps $i_{1} : A_{1} \to C, i_{2} : A_{2} \to C$ and maps $π_{1} : A_{1} \to C, π_{2} : A_{2} \to C$ such that:

$π_{1} \circ i_{1}$ and $π_{2} \circ i_{2}$ are the identity maps on $A_{1}$ and $A_{2}$ respectively.
$C$ , along with the maps $π_{1}, π_{2}$ , is a product of $A_{1}$ and $A_{2}$ . In other words, for any object $D$ with maps $f_{1} : D \to A_{1}, f_{2} : D \to A_{2}$ , there exists a unique map $g : D \to C$ such that $f_{1} = π_{1} \circ g$ and $f_{2} = π_{2} \circ g$ .
$C$ , along with the maps $i_{1}, i_{2}$ , is a coproduct of $A_{1}$ and $A_{2}$ . In other words, for any object $D$ with maps $f_{1} : A_{1} \to D, f_{2} : A_{2} \to D$ , there exists $g : C \to D$ such that $f_{1} = g \circ i_{1}, f_{2} = g \circ i_{2}$ .

For a finite collection of objects

A preadditive category that admits biproducts for finite collections of objects is termed an additive category.