Preadditive Category - Biproducts

Biproducts

Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition:

The object B is a biproduct of the objects A₁,...,A_n if and only if there are projection morphisms p_j: B → A_j and injection morphisms i_j: A_j → B, such that (i₁ o p₁) + ··· + (i_n o p_n) is the identity morphism of B, p_j o i_j is the identity morphism of A_j, and p_j o i_k is the zero morphism from A_k to A_j whenever j and k are distinct.

This biproduct is often written A₁ ⊕ ··· ⊕ A_n, borrowing the notation for the direct sum. This is because the biproduct in well known preadditive categories like Ab is the direct sum. However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense.

The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and coterminal (a nullary coproduct), it will in fact be a zero object. Indeed, the term "zero object" originated in the study of preadditive categories like Ab, where the zero object is the zero group.

A preadditive category in which every biproduct exists (including a zero object) is called additive. Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.