Finite Coproduct in Preadditive Category is Biproduct
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Theorem
Binary coproducts
Let $A$ be a preadditive category.
Let $a_1, a_2$ be objects of $A$.
Let $(a_1 \sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists.
Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the unique morphism with:
- $p_1 \circ i_1 = 1 : a_1 \to a_1$
- $p_1 \circ i_2 = 0 : a_1 \to a_2$
Let $p_2 : a_1 \sqcup a_2 \to a_2$ be the unique morphism with:
- $p_2 \circ i_1 = 0 : a_2 \to a_1$
- $p_2 \circ i_2 = 1 : a_2 \to a_2$
where $1$ is the identity morphism and $0$ is the zero morphism.
Then $(a_1 \sqcup a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.
A particular theorem is missing. In particular: arbitrary finite coproducts You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding the theorem. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{TheoremWanted}} from the code. |