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.

Also see

• Equivalence of Definitions of Binary Biproduct
• Definition:Preadditive Category
• Finite Product in Preadditive Category is Biproduct
• Finite Coproduct in Preadditive Category is Biproduct

Sources

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