Biproduct
In category theory and its applications to mathematics, a biproduct of a finite collection of objects, in a category with zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules.
Let C be a category with zero objects.
Given objects A1,...,An in C, their biproduct is an object A1 ⊕ ··· ⊕ An together with morphisms
satisfying
and such that
An empty, or nullary, product is always a terminal object in the category, and the empty coproduct is always an initial object in the category. Since our category C has a zero object, the empty biproduct exists and is isomorphic to the zero object.
In the category of abelian groups, biproducts always exist and are given by the direct sum. Note that the zero object is the trivial group.
Similarly, biproducts exist in the category of vector spaces over a field. The biproduct is again the direct sum, and the zero object is the trivial vector space.
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