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 both the tensor product and the composition-operation on morphisms are bilinear.

Beware that in discussion of differential categories and of differential linear logic such categories are often referred to as “(pre-)additive categories”, while traditionally this terminology refers to enrichment in abelian groups instead of just commutative monoids. The term “CMon-enriched symmetric monoidal category” is non-standard but used here to avoid this clash of terminology.

Definition

In detail, 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