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Let C be a preadditive category. In particular, morphisms in C can be added.
Given objects A1,...,An in C, suppose that we have:
- another object A1 ⊕ ··· ⊕ An in C (the biproduct);
- morphsims pk: A1 ⊕ ··· ⊕ An → Ak in C (the projection morphisms); and
- morphisms ik: Ak → A1 ⊕ ··· ⊕ An (the injection morphisms).
Additionally, suppose that:
- (i1 ° p1) + ··· + (in ° pn) equals the identity morphism of A1 ⊕ ··· ⊕ An;
- pk ° ik equals the identity element of Ak; and
- pk ° il is the zero morphism from Al to Ak whenever k and l are distinct.
Then A1 ⊕ ··· ⊕ An is a biproduct of A1,...,An.
Note that if we take n = 0 in the above definition, then only the first condition applies, and we have for the nullary biproduct an object O such that the identity morphism on O is equal to the zero morphism from O to itself.
Biproducts always exist in the category of abelian groups. In that category, the biproduct of several objects is simply their direct sum. The nullary biproduct is the trivial group. Biproducts exist in several other categories with direct sums, such as the category of vector spaces over a given field. But biproducts do not exist in the category of all groups; indeed, this category is not even preadditive.
If a nullary biproduct exists and all binary biproducts A1 ⊕ A2 exist, then all biproducts whatsoever must also exist; this can be proved by mathematical induction.
Biproducts in preadditive categories are always both products and coproducts in the ordinary category-theoretic sense; this is the origin of the term "biproduct". In particular, a nullary biproduct is always a zero object. Conversely, any finitary product or coproduct in a preadditive category must be a biproduct.
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