Biproduct

Definition

For two objects

Suppose ${\mathcal {C}}$ is a category and $A_{1},A_{2}\in \operatorname {Ob} {\mathcal {C}}$. 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 $\pi _{1}:A_{1}\to C,\pi _{2}:A_{2}\to C$ such that:

1. $\pi _{1}\circ i_{1}$ and $\pi _{2}\circ i_{2}$ are the identity maps on $A_{1}$ and $A_{2}$ respectively.
2. $C$, along with the maps $\pi _{1},\pi _{2}$, 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 $f_{1}=\pi _{1}\circ g$ and $f_{2}=\pi _{2}\circ g$.
3. $C$, 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.