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 morphisms. Given a finite (possibly empty) collection of objects A₁, ..., A_n in C, their biproduct is an object ${\textstyle A_{1}\oplus \dots \oplus A_{n}}$ in C together with morphisms

satisfying

and such that

If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to ${\textstyle i_{1}\circ p_{1}+\dots +i_{n}\circ p_{n}=1_{A_{1}\oplus \dots \oplus A_{n}}}$ when n > 0. 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. Thus an empty, or nullary, biproduct is always a zero object.

In the category of abelian groups, biproducts always exist and are given by the direct sum. 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.

More generally, biproducts exist in the category of modules over a ring.