# nLab CMon-enriched symmetric monoidal category

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Enriched category theory

enriched category theory

# Contents

## 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 $j,k \colon B \rightarrow C$:

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

Last revised on January 21, 2024 at 14:48:15. See the history of this page for a list of all contributions to it.