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.

Given a finite (possibly empty) collection of objects A1, ..., An in C, their biproduct is an object in C together with morphisms

  • in C (the projection morphisms)
  • (the embedding morphisms)

and such that

  • is a product for the and
  • is a coproduct for the

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.[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 exists for all pairs of objects A and B in the category C, then all finite biproducts exist, making C both a Cartesian monoidal category and a co-Cartesian monoidal category.

If the product and coproduct both exist for some pair of objects Ai, then there is a unique morphism such that

  • for

It follows that the biproduct 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 exists, then there are unique morphisms such that

  • for

To see that is now also a coproduct, and hence a biproduct, suppose we have morphisms for some object . Define Then is a morphism and .

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.

References

  1. Borceux, 4-5
  2. Borceux, 8
  3. Borceux, 7
  4. H.D. Macedo, J.N. Oliveira, Typing linear algebra: A biproduct-oriented approach, Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, ISSN 0167-6423, doi:10.1016/j.scico.2012.07.012.
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