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.
Definition.
Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects "A"1, ..., "A""n" in C, their "biproduct" is an object formula_1 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 formula_13 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.
Examples.
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.
On the other hand, biproducts do not exist in the category of groups. Here, the product is the direct product, but the coproduct is the free product.
Also, biproducts do not exist in the category of sets. For, the product is given by the Cartesian product, whereas the coproduct is given by the disjoint union. This category does not have a zero object.
Block matrix algebra relies upon biproducts in categories of matrices.
Properties.
If the biproduct formula_14 exists for all pairs of objects "A" and "B" in the category C, and C has a zero object, then all finite biproducts exist, making C both a Cartesian monoidal category and a co-Cartesian monoidal category.
If the product formula_15 and coproduct formula_16 both exist for some pair of objects "A"1, "A"2 then there is a unique morphism formula_17 such that
It follows that the biproduct formula_21 exists if and only if "f" is an isomorphism.
If C is a preadditive category, then every finite product is a biproduct, and every finite coproduct is a biproduct. For example, if formula_15 exists, then there are unique morphisms formula_23 such that
To see that formula_15 is now also a coproduct, and hence a biproduct, suppose we have morphisms formula_28 for some object formula_29. Define formula_30 Then formula_31 is a morphism from formula_15 to formula_29, and formula_34 for formula_35.
In this case we always have
An additive category is a preadditive category in which all finite biproducts exist. In particular, biproducts always exist in abelian categories.