With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
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 $j,k \colon B \rightarrow C$:
Last revised on January 21, 2024 at 14:48:15. See the history of this page for a list of all contributions to it.