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

-tuply monoidal -categories

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Monoidal categories

monoidal categories

With symmetry

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

kk-tuply monoidal (n,r)(n,r)-categories

Idea

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

Definition

A kk-tuply monoidal (n,r)(n,r)-category is a pointed \infty-category (which you may interpret as weakly or strictly as you like) such that: * any two parallel jj-morphisms are equivalent, for j<kj \lt k; * any jj-morphism is an equivalence, for j>r+kj \gt r + k; * any two parallel jj-morphisms are equivalent, for j>n+kj \gt n + k.

Keep in mind that one usually relabels the jj-morphisms as (jk)(j-k)-morphisms, which explains the usage of r+kr + k and n+kn + k instead of rr and nn. As explained below, we may assume that n1n \geq -1, 1rn+1-1 \leq r \leq n + 1, 0kn+20 \leq k \leq n + 2, and (if convenient) r+k0r + k \geq 0.

To interpret this correctly for low values of jj, assume that all objects (00-morphisms) in a given \infty-category are parallel, which leads one to speak of the two (1)(-1)-morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any jj-morphism is an equivalence for j<1j \lt 1, so if r+k=0r + k = 0, then the condition is satisfied for any smaller value of r+kr + k. Thus, we may assume that r+k0r + k \geq 0. Similarly, since there is a chosen object (the basepoint), any parallel jj-morphisms are equivalent for j<1j \lt 1,

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

If any two parallel jj-morphisms are equivalent, then any jj-morphism between equivalent (j1)(j-1)-morphisms is an equivalence (being parallel to an equivalence for j>0j \gt 0 and automatically for j<1j \lt 1). Accordingly, any kk-tuply monoidal (n,0)(n,0)-category is automatically also a kk-tuply monoidal (n,r)(n,r)-category for any r<0r \lt 0, and any kk-tuply monoidal (n,r)(n,r)-category for r>n+1r \gt n + 1 is also a kk-tuply monoidal (n,n+1)(n,n+1)-category. Thus, we don't need r<1r \lt -1 or r>n+1r \gt n + 1.

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

Special cases

A 00-tuply monoidal (n,r)(n,r)-category is simply a pointed (n,r)(n,r)-category. The restriction that r+k0r + k \geq 0 becomes that r0r \geq 0. This is why (n,r)(n,r)-categories use 0rn+10 \leq r \leq n + 1 rather than the restriction on rr given before.

A kk-tuply monoidal (n,0)(n,0)-category is a kk-tuply monoidal nn-groupoid. A kk-tuply monoidal (n,1)(n,-1)-category is a kk-tuply groupal nn-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)(n,r)-categories. (But if we moved to a 4D table that required all jj-morphisms to be equivalences for sufficiently low values of jj, then groupal categories would appear there.)

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

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

Last revised on October 26, 2012 at 04:34:58. See the history of this page for a list of all contributions to it.