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 in (Baez-Dolan 97) and developed by (Makkai) and others: opetopic ω-categories.
A syntactic formalization of opetopic ω-categories in the variant by Palm is opetopic type theory (Finster 12).
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, John Power, On weak higher-dimensional categories I, II Jour. Pure Appl. Alg. 157 (2001), 221–277 (journal, ps.gz files)
Michael Makkai, The multitopic $\omega$-category of all multitopic $\omega$-categories. (web)
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 higher dimensional string diagram-notation for opetopes was introduced (as “zoom complexes” in section 1.1) in
Animated exposition of this higher-dimensional string-diagram notation is in
Eric Finster, Opetopic Diagrams 1 - Basics (video)
Eric Finster, Opetopic Diagrams 2 - Geometry (video)
The variant of Palm opetopic omega-categories is due to
A syntactic formalization of opetopic omega-categories in terms of opetopic type theory is in