Definition:Binary Biproduct

Definition
Let $A$ be a category.

Let $a_1,a_2$ be objects of $A$.

Definition 1
A biproduct of $a_1$ and $a_2$ is an ordered tuple $(a_1\oplus a_2, p_1, p_2, i_1, i_2)$ such that:
 * $(a_1\oplus a_2, p_1, p_2)$ is a binary product
 * $(a_1\oplus a_2, i_1, i_2)$ is a binary coproduct

Definition 2: for categories with zero morphisms
Let $A$ be a category with zero morphisms.

Then $a_1$ and $a_2$ are said to have a biproduct :
 * they have a coproduct $(a_1 \sqcup a_2, j_1, j_2)$ and a product $(a_1 \times a_2, q_1, q_2)$
 * the canonical mapping $r : a_1 \sqcup a_2 \to a_1 \times a_2$ is an isomorphism

in which case:
 * $(a_1 \sqcup a_2, j_1, j_2, q_1 \circ r, q_2 \circ r)$ and
 * $(a_1 \times a_2, r \circ j_1, r \circ j_2, q_1, q_2)$

are biproducts of $a_1$ and $a_2$.

Definition 3: for preadditive categories
Let $A$ be a preadditive category.

A biproduct of $a_1$ and $a_2$ is an ordered tuple $(a_1\oplus a_2, p_1, p_2, i_1, i_2)$ where:
 * $a \oplus a_2$ is an object of $A$
 * $i_1 : a_1 \to a_1 \oplus a_2$
 * $i_2 : a_1 \to a_1 \oplus a_2$
 * $p_1 : a_1 \oplus a_2 \to a_2$
 * $p_2 : a_1 \oplus a_2 \to a_2$

are morphisms such that:
 * $p_1 \circ i_1 = 1_{a_1}$
 * $p_2 \circ i_2 = 1_{a_2}$
 * $i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1\oplus a_2}$

where $1$ denotes the identity morphism.

Definition 4: for preadditive categories
Let $A$ be a preadditive category.

A biproduct of $a_1$ and $a_2$ is an ordered tuple $(a_1\oplus a_2, p_1, p_2, i_1, i_2)$ where:
 * $a \oplus a_2$ is an object of $A$
 * $i_1 : a_1 \to a_1 \oplus a_2$
 * $i_2 : a_1 \to a_1 \oplus a_2$
 * $p_1 : a_1 \oplus a_2 \to a_2$
 * $p_2 : a_1 \oplus a_2 \to a_2$

are morphisms such that:
 * $p_1 \circ i_1 = 1_{a_1}$
 * $p_2 \circ i_2 = 1_{a_2}$
 * $p_1 \circ i_2 = 0_{a_1}$
 * $p_2 \circ i_1 = 0_{a_2}$
 * $i_1 \circ p_1 + i_2 \circ p_2 = 1_{a_1\oplus a_2}$

where:
 * $1$ denotes the identity morphism
 * $0$ denotes the zero morphism.

Also see

 * Equivalence of Definitions of Binary Biproduct
 * Finite Product in Preadditive Category is Biproduct
 * Finite Coproduct in Preadditive Category is Biproduct