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.

Let C be a category with zero morphisms. Given a finite (possibly empty) collection of objects A₁, ..., A_n in C, their biproduct is an object ${\textstyle A_{1}\oplus \dots \oplus A_{n}}$ in C together with morphisms

• ${\textstyle p_{k}\!:A_{1}\oplus \dots \oplus A_{n}\to A_{k}}$ in C (the projection morphisms)
• ${\textstyle i_{k}\!:A_{k}\to A_{1}\oplus \dots \oplus A_{n}}$ (the embedding morphisms)

satisfying

• ${\textstyle p_{k}\circ i_{k}=1_{A_{k}}}$, the identity morphism of ${\displaystyle A_{k},}$ and
• ${\textstyle p_{l}\circ i_{k}=0}$, the zero morphism ${\displaystyle A_{k}\to A_{l},}$ for ${\displaystyle k\neq l,}$

and such that

• ${\textstyle \left(A_{1}\oplus \dots \oplus A_{n},p_{k}\right)}$ is a product for the ${\textstyle A_{k},}$ and
• ${\textstyle \left(A_{1}\oplus \dots \oplus A_{n},i_{k}\right)}$ is a coproduct for the ${\textstyle A_{k}.}$

If C is preadditive and the first two conditions hold, then each of the last two conditions is equivalent to ${\textstyle i_{1}\circ p_{1}+\dots +i_{n}\circ p_{n}=1_{A_{1}\oplus \dots \oplus A_{n}}}$ 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.

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]

If the biproduct ${\textstyle A\oplus B}$ 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 ${\textstyle A_{1}\times A_{2}}$ and coproduct ${\textstyle A_{1}\coprod A_{2}}$ both exist for some pair of objects A₁, A₂ then there is a unique morphism ${\textstyle f:A_{1}\coprod A_{2}\to A_{1}\times A_{2}}$ such that

• ${\displaystyle p_{k}\circ f\circ i_{k}=1_{A_{k}},\ (k=1,2)}$
• ${\displaystyle p_{l}\circ f\circ i_{k}=0}$ for ${\textstyle k\neq l.}$^{[clarification needed]}

It follows that the biproduct ${\textstyle A_{1}\oplus A_{2}}$ 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 ${\textstyle A_{1}\times A_{2}}$ exists, then there are unique morphisms ${\textstyle i_{k}:A_{k}\to A_{1}\times A_{2}}$ such that

• ${\displaystyle p_{k}\circ i_{k}=1_{A_{k}},\ (k=1,2)}$
• ${\displaystyle p_{l}\circ i_{k}=0}$ for ${\textstyle k\neq l.}$

To see that ${\textstyle A_{1}\times A_{2}}$ is now also a coproduct, and hence a biproduct, suppose we have morphisms ${\textstyle f_{k}:A_{k}\to X,\ k=1,2}$ for some object ${\textstyle X}$. Define ${\textstyle f:=f_{1}\circ p_{1}+f_{2}\circ p_{2}.}$ Then ${\textstyle f}$ is a morphism from ${\textstyle A_{1}\times A_{2}}$ to ${\textstyle X}$, and ${\textstyle f\circ i_{k}=f_{k}}$ for ${\textstyle k=1,2}$.

In this case we always have

• ${\textstyle i_{1}\circ p_{1}+i_{2}\circ p_{2}=1_{A_{1}\times A_{2}}.}$

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

• Borceux, Francis (2008). Handbook of Categorical Algebra 2: Categories and Structures. Cambridge University Press. ISBN 978-0-521-06122-3.^{: Section 1.2}