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(n × k)-category

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Higher category theory

higher category theory

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Contents

Idea

An (n×k)(n \times k)-category (read “n-by-k category”) is an n-category internal to a kk-category. 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.

Examples

Relationships

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).

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 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 August 12, 2019 at 13:49:36. See the history of this page for a list of all contributions to it.