# Definition:Binary Biproduct

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## 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 $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ such that:

- $\tuple {a_1 \oplus a_2, p_1, p_2}$ is a binary product
- $\tuple {a_1 \oplus a_2, i_1, i_2}$ 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 $\tuple {a_1 \sqcup a_2, j_1, j_2}$ and a product $\tuple {a_1 \times a_2, q_1, q_2}$
- the canonical mapping $r: a_1 \sqcup a_2 \to a_1 \times a_2$ is an isomorphism

in which case:

- $\tuple {a_1 \sqcup a_2, j_1, j_2, q_1 \circ r, q_2 \circ r}$
- $\tuple {a_1 \times a_2, r \circ j_1, r \circ j_2, q_1, q_2}$

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 $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ 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 $\tuple {a_1 \oplus a_2, p_1, p_2, i_1, i_2}$ 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.