category object in an (∞,1)-category, groupoid object
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:
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.
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).
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 $n, k \leq 3$.
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.