In category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category.

Definition

Let C be a preadditive category. In particular, morphisms in C can be added.

Given objects A₁,...,A_n in C, suppose that we have:

• another object A₁ ⊕ ··· ⊕ A_n in C (the biproduct);
• morphsims p_k: A₁ ⊕ ··· ⊕ A_n → A_k in C (the projection morphisms); and
• morphisms i_k: A_k → A₁ ⊕ ··· ⊕ A_n (the injection morphisms).

Additionally, suppose that:

• (i_{1 °} p₁) + ··· + (i_{n °} p_n) equals the identity morphism of A₁ ⊕ ··· ⊕ A_n;
• p_{k °} i_k equals the identity element of A_k; and
• p_{k °} i_l is the zero morphism from A_l to A_k whenever k and l are distinct.

Then A₁ ⊕ ··· ⊕ A_n is a biproduct of A₁,...,A_n.

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.

Examples

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.

Properties

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.

An additive category is a preadditive category in which every biproduct exists. In particular, biproducts always exist in abelian categories.