# nLab k-tuply monoidal (n,r)-category

-tuply monoidal -categories

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### 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

# $k$-tuply monoidal $(n,r)$-categories

## Idea

Two important periodic tables are the table of $k$-tuply monoidal $n$-categories and the table of $(n,r)$-categories. These can actually be combined into a single 3D table, which surprisingly also includes $k$-tuply groupal $n$-groupoids.

## Definition

A $k$-tuply monoidal $(n,r)$-category is a pointed $\infty$-category (which you may interpret as weakly or strictly as you like) such that:

• any two parallel $j$-morphisms are equivalent, for $j \lt k$;
• any $j$-morphism is an equivalence, for $j \gt r + k$;
• any two parallel $j$-morphisms are equivalent, for $j \gt n + k$.

Keep in mind that one usually relabels the $j$-morphisms as $(j-k)$-morphisms, which explains the usage of $r + k$ and $n + k$ instead of $r$ and $n$. As explained below, we may assume that $n \geq -1$, $-1 \leq r \leq n + 1$, $0 \leq k \leq n + 2$, and (if convenient) $r + k \geq 0$.

To interpret this correctly for low values of $j$, assume that all objects ($0$-morphisms) in a given $\infty$-category are parallel, which leads one to speak of the two $(-1)$-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any $j$-morphism is an equivalence for $j \lt 1$, so if $r + k = 0$, then the condition is satisfied for any smaller value of $r + k$. Thus, we may assume that $r + k \geq 0$. Similarly, since there is a chosen object (the basepoint), any parallel $j$-morphisms are equivalent for $j \lt 1$,

The conditions that $j \lt k$ and that $j \gt n + k$ will overlap if $n \lt - 1$, so we don't use such values of $n$. In other words, any $k$-tuply monoidal $(-1,r)$-category is also a $k$-tuply monoidal $(n,r)$-category for any $n \lt - 1$.

If any two parallel $j$-morphisms are equivalent, then any $j$-morphism between equivalent $(j-1)$-morphisms is an equivalence (being parallel to an equivalence for $j \gt 0$ and automatically for $j \lt 1$). Accordingly, any $k$-tuply monoidal $(n,0)$-category is automatically also a $k$-tuply monoidal $(n,r)$-category for any $r \lt 0$, and any $k$-tuply monoidal $(n,r)$-category for $r \gt n + 1$ is also a $k$-tuply monoidal $(n,n+1)$-category. Thus, we don't need $r \lt -1$ or $r \gt n + 1$.

According to the stabilisation hypothesis, every $k$-tuply monoidal $(n,r)$-category for $k \gt n + 2$ may be reinterpreted as an $(n+2)$-tuply monoidal $(n,r)$-category. Unlike the other restrictions on values of $n, r, k$, this one is not trivial.

## Special cases

A $0$-tuply monoidal $(n,r)$-category is simply a pointed $(n,r)$-category. The restriction that $r + k \geq 0$ becomes that $r \geq 0$. This is why $(n,r)$-categories use $0 \leq r \leq n + 1$ rather than the restriction on $r$ given before.

A $k$-tuply monoidal $(n,0)$-category is a $k$-tuply monoidal $n$-groupoid. A $k$-tuply monoidal $(n,-1)$-category is a $k$-tuply groupal $n$-groupoid. This is why groupal categories? don't come up much; the progression from monoidal categories to monoidal groupoids to groupal groupoids? is a straight line up one column of the periodic table of monoidal? $(n,r)$-categories. (But if we moved to a 4D table that required all $j$-morphisms to be equivalences for sufficiently low values of $j$, then groupal categories would appear there.)

A $k$-tuply monoidal $(n,n)$-category is simply a $k$-tuply monoidal $n$-category. A $k$-tuply monoidal $(n,n+1)$-category is a $k$-tuply monoidal $(n+1)$-poset. Note that a $k$-tuply monoidal $\infty$-category and a $k$-tuply monoidal $\infty$-poset are the same thing.

A stably monoidal $(n,r)$-category, or symmetric monoidal $(n,r)$-category, is an $(n+2)$-tuply monoidal $(n,r)$-category. Although the general definition above won't give it, there is a notion of stably monoidal $(\infty,r)$-category, basically an $(\infty,r)$-category that can be made $k$-tuply monoidal for any value of $k$ in a consistent way.

Last revised on March 4, 2023 at 21:21:49. See the history of this page for a list of all contributions to it.