Biproduct

Examples

In the category of abelian groups, biproducts always exist and are given by the direct sum. 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. This category does not have a zero object.

Block matrix algebra relies upon biproducts in categories of matrices.

Properties

If the biproduct $A\oplus B$ 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 $A_1\times A_2$ and coproduct $A_1\coprod A_2$ both exist for some pair of objects A₁, A₂ then there is a unique morphism $f:A_1\coprod A_2\to A_1\times A_2$ 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 $k\ne l.$

It follows that the biproduct $A_1\oplus A_2$ 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 $A_1\times A_2$ exists, then there are unique morphisms $i_k:A_k\to A_1\times A_2$ such that

• ${\displaystyle p_k\circ i_k=1_{A_k},(k=1,2)}$
• ${\displaystyle p_l\circ i_k=0}$ for $k\ne l.$

To see that $A_1\times A_2$ is now also a coproduct, and hence a biproduct, suppose we have morphisms $f_k:A_k\to X,k=1,2$ for some object $X$. Define $f:=f_1\circ p_1+f_2\circ p_2.$ Then $f$ is a morphism from $A_1\times A_2$ to $X$, and $f\circ i_k=f_k$ for $k=1,2$.

In this case we always have

• $i_1\circ p_1+i_2\circ p_2=1_{A_1\times A_2}.$

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