Biproduct Explained

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

• $p\_k\; \backslash !:\; A\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; A\_n\; \backslash to\; A\_k$ in C (the projection morphisms)
• $i\_k\; \backslash !:\; A\_k\; \backslash to\; A\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; A\_n$ (the embedding morphisms)

satisfying

• $p\_k\; \backslash circ\; i\_k\; =\; 1\_$, the identity morphism of

A _{k,} andA _{k} \to A _{l,} for k ≠ l, and such that

• $\backslash left(A\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; A\_n,\; p\_k\; \backslash right)$ is a product for the $A\_k,$ and
• $\backslash left(A\_1\; \backslash oplus\; \backslash dots\; \backslash oplus\; A\_n,\; i\_k\; \backslash right)$ is a coproduct for the $A\_k.$

If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to $i\_1\; \backslash circ\; p\_1\; +\; \backslash dots\; +\; i\_n\backslash circ\; p\_n\; =\; 1\_$ when n > 0.^[2] 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.^[3] 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.^[4] 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.^[5]

Properties

If the biproduct $A\; \backslash oplus\; B$ 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 $A\_1\; \backslash times\; A\_2$ and coproduct $A\_1\; \backslash coprod\; A\_2$ both exist for some pair of objects A₁, A₂ then there is a unique morphism $f:\; A\_1\; \backslash coprod\; A\_2\; \backslash to\; A\_1\; \backslash times\; A\_2$ such that

p _{k} \circ f \circ i _{k} = 1 _{A _{k}} , (k = 1 , 2 ) p _{l} \circ f \circ i _{k} = 0 for $k\; \backslash neq\; l.$

It follows that the biproduct $A\_1\; \backslash oplus\; A\_2$ 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 $A\_1\; \backslash times\; A\_2$ exists, then there are unique morphisms $i\_k:\; A\_k\; \backslash to\; A\_1\; \backslash times\; A\_2$ such that

p _{k} \circ i _{k} = 1 _{A _{k}} , (k = 1 , 2 ) p _{l} \circ i _{k} = 0 for $k\; \backslash neq\; l.$

To see that $A\_1\; \backslash times\; A\_2$ is now also a coproduct, and hence a biproduct, suppose we have morphisms $f\_k:\; A\_k\; \backslash to\; X,\backslash \; k=1,2$ for some object $X$. Define $f\; :=\; f\_1\; \backslash circ\; p\_1\; +\; f\_2\; \backslash circ\; p\_2.$ Then $f$ is a morphism from $A\_1\; \backslash times\; A\_2$ to $X$, and $f\; \backslash circ\; i\_k\; =\; f\_k$ for $k\; =\; 1,\; 2$.

In this case we always have

• $i\_1\; \backslash circ\; p\_1\; +\; i\_2\; \backslash circ\; p\_2\; =\; 1\_.$

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

References

• Book: Borceux. Francis. Handbook of Categorical Algebra 2: Categories and Structures. 2008. Cambridge University Press. 978-0-521-06122-3.

Notes and References

1. Borceux, 4-5
2. Saunders Mac Lane - Categories for the Working Mathematician, Second Edition, page 194.
3. Borceux, 8
4. Borceux, 7
5. 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,, .