nLab CMon-enriched monoidal category

Idea

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

Definition

In details, a CMon-enriched monoidal category is a 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 $j,k \colon B \rightarrow C$:

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

Properties

Proposition

If a CMon-enriched monoidal category possesses the binary products, then they are biproducts and the category is thus semiadditive. The same goes if it possesses the binary coproducts.

Proposition

If a CMon-enriched monoidal category possesses a terminal object, then it is a zero object. The same goes if it possesses an initial object.

Examples

Proposition

Every semiadditive monoidal category is a CMon-enriched monoidal category.

Related entries

• CMon-enriched symmetric monoidal category