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

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 $\mathrm{\infty }$-category (which you may interpret as weakly or strictly as you like) such that:

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

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 $n\ge -1$, $-1\le r\le n+1$, $0\le k\le n+2$, and (if convenient) $r+k\ge 0$.

To interpret this correctly for low values of $j$, assume that all objects ($0$-morphisms) in a given $\mathrm{\infty }$-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 $j<1$, so if $r+k=0$, then the condition is satisfied for any smaller value of $r+k$. Thus, we may assume that $r+k\ge 0$. Similarly, since there is a chosen object (the basepoint), any parallel $j$-morphisms are equivalent for $j<1$,

The conditions that $j<k$ and that $j>n+k$ will overlap if $n<-1$, 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 $n<-1$.

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 $j>0$ and automatically for $j<1$). Accordingly, any $k$-tuply monoidal $(n,0)$-category is automatically also a $k$-tuply monoidal $(n,r)$-category for any $r<0$, and any $k$-tuply monoidal $(n,r)$-category for $r>n+1$ is also a $k$-tuply monoidal $(n,n+1)$-category. Thus, we don't need $r<-1$ or $r>n+1$.

According to the stabilisation hypothesis, every $k$-tuply monoidal $(n,r)$-category for $k>n+2$ 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 $r+k\ge 0$ becomes that $r\ge 0$. This is why $(n,r)$-categories use $0\le r\le n+1$ 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 $\mathrm{\infty }$-category and a $k$-tuply monoidal $\mathrm{\infty }$-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 $(\mathrm{\infty },r)$-category, basically an $(\mathrm{\infty },r)$-category that can be made $k$-tuply monoidal for any value of $k$ in a consistent way.

Related concepts

• stabilization hypothesis