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.
Contents
Definition
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.
Examples
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.
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. Note also that this category does not have a zero object.
Block matrix algebra relies upon biproducts in categories of matrices.
Properties
If the biproduct A ⊕ B exists for all pairs of objects A and B in the category C, then all finite biproducts exist.
If the product A1 × A2 and coproduct A1 ∐ A2 both exist for some pair of objects Ai, then there is a unique morphism f: A1 ∐ A2 → A1 × A2 such that
It follows that the biproduct A1 ⊕ A2 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 A1 × A2 exists, then there are unique morphisms ik: Ak → A1 × A2 such that
To see that A1 × A2 is now also a coproduct, and hence a biproduct, suppose we have morphisms fk: Ak → X for some object X. Define f := f1 ∘ p1 + f2 ∘ p2. Then f: A1 × A2 → X is a morphism and f ∘ ik = fk.
Note also that in this case we always have
An additive category is a preadditive category in which all finite biproduct exist. In particular, biproducts always exist in abelian categories.