recollection - m-semiadditive higher categories

This definition appears in upcoming work of Scheimbauer-Walde

Idea A categorification of a commutative monoid is a semiadditive -category.

A categorification of an $m$-commutative monoid (in the sense of Harpaz) is an $m$-semiadditive category.

For $n\ge 2$, one can make the following recursive definition

Definition: An $(\infty ,n)$-category $C$ is $m$-semiadditive if

• the underlying $(\infty ,1)$-category of $m$-semi-additive AND
• the hom-$(\infty ,n-1)$-categories $Ma{p}_C(X,Y)$ are $m$-semiadditive $(\infty ,n-1)$-categories

Observe that by construction, at each $n$, semi-additivity is a property rather than extra structure.

Remark At $n=1$, the second line in the condition doesn’t quite make sense: it doesn’t mean anything for the mapping anima $Map(X,Y)$ to be a “semi-additive groupoid”.

We can rephrase the definition in a more uniform way that DOES extend down though, by saying an $n$-category is $m$-semiadditive if its underlying $1$-category is, and additionally it is tensored in $m$-semiadditive $(n-1)$-categories. A priori this looks like extra data of a categorical action, but we see at the base case $n=1$, this is asking for a category to be tensored over $CMon$, and this extra data in fact is property, because $CMon$ is a mode.