homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
Opetopes are one of the geometric shapes of cells in the approach to the higher category theory of n-categories and ω-categories put forward by John Baez and James Dolan.
An overview is in chapter 4 of
and in chapter 7 of
Opetopes were introduced here:
Some mistakes were corrected in subsequent papers:
Eugenia Cheng, The category of opetopes and the category of opetopic sets, Th. Appl. Cat. 11 (2003), 353–374. arXiv)
Tom Leinster, Structures in higher-dimensional category theory. (arXiv)
Makkai and collaborators introduced a slight variation they called ‘multitopes’:
Claudio Hermida, Michael Makkai, and J. Power: On weak higher-dimensional categories I, II. Jour. Pure Appl. Alg. 157 (2001), 221–277.
Michael Makkai, The multitopic $\omega$-category of all multitopic $\omega$-categories. (online)
Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic $n$-categories:
Eugenia Cheng, Weak $n$-categories: opetopic and multitopic foundations, Jour. Pure Appl. Alg. 186 (2004), 109–137.(arXiv)
Eugenia Cheng, Weak $n$-categories: comparing opetopic foundations, Jour. Pure Appl. Alg. 186 (2004), 219–231. (arXiv)
She has also shown that opetopic bicategories are “the same” as the ordinary kind:
A formalization of the definition of opetotes in type theory is in