nLab CMon-enriched symmetric monoidal category

Idea

A CMon-enriched symmetric monoidal category is a symmetric monoidal category such that each hom-set is a commutative monoid and the tensor product on morphisms and the composition are bilinear.

This notion is often referred to as additive category or pre-additive category in works on differential categories or differential linear logic. However these terms classicaly assume more structure, the most obvious being an enrichment over abelian groups and not only commutative monoids. This is typical of the attitude in theoretical computer science where one often doesn’t assume the presence of negative numbers. In order to prevent confusion, we prefer using the name defended here.

Definition

In details, a CMon-enriched symmetric monoidal category is a symmetric monoidal category such that each hom-set $\mathcal{C}[A,B]$ is a commutative monoid (we write $f + g$ for the sum of two morphisms $f,g \colon A \rightarrow B$ and $0$ for the zero $A \rightarrow B$), such that for every $f,g \colon A \rightarrow B$, $h,i \colon C \rightarrow D$ and $j,k \colon B \rightarrow C$:

• $(f+g)\otimes h = f \otimes h + g \otimes h$
• $f \otimes (h+i) = f \otimes h + f \otimes i$
• $0 \otimes f = f \otimes 0 = 0$
• $(f+g);j = f;j + g;j$
• $f;(j+k) = f;j + f;k$
• $0;f = f;0 = 0$

Related entries

• CMon-enriched monoidal category