nLab (n × k)-category



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations

Internal categories



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

For example:

  • A (1×0)(1 \times 0)-category, as well as a (0×1)(0 \times 1)-category, is precisely a category. More generally, (n×0)(n\times 0)-categories and (0×n)(0\times n)-categories are precisely nn-categories.
  • A (1×1)(1 \times 1)-category is precisely a double category (either strict or weak).
  • Generalizing to a 3rd axis, a (1×1×1)(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×1)(n \times 1)-category is what Batanin calls a monoidal n-globular category?.

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

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



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

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

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

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


  • 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 nn-categories , Advances in Mathematics 136 (1998), no. 1, 39–103.

The following paper contains some discussion on the relationship between various (weak) (n×k)(n \times k)-categories for n,k3n, k \leq 3.

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

Last revised on March 29, 2024 at 23:26:36. See the history of this page for a list of all contributions to it.