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 $(a_1\oplus a_2, p_1, p_2, i_1, i_2)$ 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.
