Semiadditve Category

A category $\mathcal{C}$ is semiadditive if it has all Finite Biproducts.

Examples

• The Category of Commutative Monoids
• The Category of Abelian Groups
• The Category of Relations

Non-Examples

• The Category of Groups and the Category of Monoids

  There are a couple of ways to understand this failure. The most obvious way is to simply compute what the Products and Coproducts are in the relevant categories. For Groups, the product is given by the Cartesian Product of Groups, but the coproduct is the Free Product of Groups.

Properties

• Every semiadditive category is enriched over the Category of Commutative Monoids.

  Finite biproducts imply a the existence of a Zero Object, which in turn implies that every hom set ${{X}\to{Y}}$ contains a map $0:{{X}\to{Y}}$.

  Moreover, we can add two parallel morphisms $f,g:{{X}\to{Y}}$ like so:

  $\par f+g=X\xrightarrow{\Delta}X\oplus X\xrightarrow{f\oplus g}Y\oplus Y \xrightarrow{\nabla}Y\par$

  If we unfold what this does in a Concrete Category, we will note that this is essentially performing pointwise addition; we duplicate the input argument, apply $f$ and $g$ to it separately, and then add the results together using the universal map out of the coproduct, which in semiadditive categories typically takes a pair, applies both maps separately, and then sums them.

• The existence of Dual Objects tends to imply additivity; for instance, every Compact Closed Category with Products or Coproducts is semiadditive. This is a sweeping generalization of the fact that if you have $\lnot$, then $\land$ and $\lor$ are mutually interdefinable.