homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
category object in an (∞,1)-category, groupoid object
An -category (read “n-by-k category”) is an n-category internal to the -category of -categories. The term is “generic” in that it does not specify the level of strictness of the -category and the -category.
For example:
An -category has kinds of cells.
Under suitable fibrancy conditions, a -category will have an underlying -category (where here, is to be read arithmetically, rather than simply as notation). Fibrant -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 to . For instance:
A sufficiently strict -category canonically gives rise to a -category. (Cor. 3.11 in DH10)
Any double category (i.e. a -category) has an underlying 2-category.
A sufficiantly fibrant -category has an underlying tricategory (i.e. -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 -categories , Advances in Mathematics 136 (1998), no. 1, 39–103.
The following paper contains some discussion on the relationship between various (weak) -categories for .
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.