nLab (n × k)-category

Idea

An $(n \times k)$-category (read “n-by-k category”) is an n-category internal to the $(k+1)$-category of $k$-categories. The term is “generic” in that it does not specify the level of strictness of the $n$-category and the $k$-category.

For example:

• A $(1 \times 0)$-category, as well as a $(0 \times 1)$-category, is precisely a category. More generally, $(n\times 0)$-categories and $(0\times n)$-categories are precisely $n$-categories.
• A $(1 \times 1)$-category is precisely a double category (either strict or weak).
• Generalizing to a 3rd axis, a $(1 \times 1 \times 1)$-category is precisely a triple category, that is, a category internal to (categories internal to categories), i.e. a catgory internal to double categories, or a double category internal to categories — which again could be strict or weak.
• An $(n \times 1)$-category is what Batanin calls a monoidal n-globular category?.

An $(n \times k)$-category has $(n + 1)(k + 1)$ kinds of cells.

Under suitable fibrancy conditions, a $(n \times k)$-category will have an underlying $(n + k)$-category (where here, $n + k$ is to be read arithmetically, rather than simply as notation). Fibrant $(1 \times 1)$-categories are known as framed bicategories.

Examples

• Commutative rings, algebras and modules form a symmetric monoidal $(2 \times 1)$-category.
• Conformal nets form a symmetric monoidal $(2 \times 1)$-category.

Relationships

At least in some cases, if the structure is sufficiently strict or sufficiently fibrant, we can shift cells from $k$ to $n$. For instance:

• A sufficiently strict $(1 \times 2)$-category canonically gives rise to a $(2 \times 1)$-category. (Cor. 3.11 in DH10)

• Any double category (i.e. a $(1\times 1)$-category) has an underlying 2-category.

• A sufficiantly fibrant $(2\times 1)$-category has an underlying tricategory (i.e. $(3\times 0)$-category).

Related concepts

• internal category

• higher category

• n-fold category

• (n,r)-category

References

• Mike Shulman, Constructing symmetric monoidal bicategories, arXiv preprint arXiv:1004.0993 (2010)

• Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak $n$-categories , Advances in Mathematics 136 (1998), no. 1, 39–103.

The following paper contains some discussion on the relationship between various (weak) $(n \times k)$-categories for $n, k \leq 3$.

• Chris Douglas, and Andre Henriques, Internal bicategories (arXiv:1206.4284) (2012)

There is some discussion on this n-Category Café post as well as this one.