# 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 if and only if:

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.