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.^[1] The biproduct is a generalization of finite direct sums of modules.

Definition

Let C be a category with zero objects.

Given objects A₁,...,A_n in C, their biproduct is an object A₁ ⊕ ··· ⊕ A_n together with morphisms

• p_k: A₁ ⊕ ··· ⊕ A_n → A_k in C (the projection morphisms)
• i_k: A_k → A₁ ⊕ ··· ⊕ A_n (the injection morphisms)

satisfying

• p_k ∘ i_k = 1_Ak, the identity morphism of A_k
• p_l ∘ i_k = 0, the zero morphism A_k → A_l, for k ≠ l.

and such that

• (A₁ ⊕ ··· ⊕ A_n,p_k) is a product for the A_k
• (A₁ ⊕ ··· ⊕ A_n,i_k) is a coproduct for the A_k.

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.^[2] 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.^[3] 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.^[4]

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 A_i, then there is a unique morphism f: A₁ ∐ A₂ → A₁ × A₂ such that

• p_k ∘ f ∘ i_k = 1_Ak
• p_l ∘ f ∘ i_k = 0 for k ≠ l.

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 i_k: A_k → A₁ × A₂ such that

• p_k ∘ i_k = 1_Ak
• p_l ∘ i_k = 0 for k ≠ l.

To see that A₁ × A₂ is now also a coproduct, and hence a biproduct, suppose we have morphisms f_k: A_k → X for some object X. Define f := f₁ ∘ p₁ + f₂ ∘ p₂. Then f: A₁ × A₂ → X is a morphism and f ∘ i_k = f_k.

Note also that in this case we always have

• i₁ ∘ p₁ + i₂ ∘ p₂ = 1_{A1 × A2}.

An additive category is a preadditive category in which all finite biproduct exist. In particular, biproducts always exist in abelian categories.

References

• Borceux, Francis (2008). Handbook of Categorical Algebra. Cambridge University Press. ISBN 978-0-521-06122-3.