In higher category theory, a concept of opetopic omega-category is one of weak omega-category modeled on opetopic shapes.
There are various flavors of the definition. Typically they all say that an opetopic omega-category is an opetopic set equipped with certain structure and property.
(Hermida-Makkai-Power 01, Makkai)
The actual definition of opetopic $\omega$-categories, called “multitopic $\omega$-categories” here, appears on p.57 of (Makkai).
In the definition by (Palm) of opetopic $\omega$-category, the extra structure on an opetopic set is that some cells are labeled as “thin” (as equivalences) and the extra property are two classes of higher dimensional horn-filler conditions:
given a horn (“niche”, “nook”) obtained from an opetope by discarding the codimension-0 interior and the outgoing codimension-1 face, then there is a filler opetope whose codimension-0 interior is marked as an equivalence. The filling outgoing codimension-1 face is marked an equivalence if all the other codimension-1 faces are.
given a horn (“niche”, “nook”) obtained from an opetope by discarding the codimension-0 interior and an incoming codimension-1 face such that all the remaining codimension-1 faces are marked as equivalences, then there exists a filler both whose codimension-1 and codimension-0 cell are marked as equivalences.
This definition has been given a syntactic formulation by (Finster 12) in terms of opetopic type theory.
However, there is no saturation condition in this definition.
An overview is in chapter 4 of
and in chapter 7 of
Opetopes were introduced here:
III: $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145–206. (arXiv:q-alg/9702014)
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:0304284)
Tom Leinster, Structures in higher-dimensional
category theory_, (arXiv:0109021)
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, pdf)
Michael Makkai, The multitopic $\omega$-category of all multitopic $\omega$-categories.
Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic $n$-categories:
Eugenia Cheng, Weak $n$-categories: comparing opetopic foundations, Jour. Pure Appl. Alg. 186 (2004), 219–231.
Eugenia Cheng, Weak $n$-categories: opetopic and multitopic
foundations_, Jour. Pure Appl. Alg._186 (2004), 109–137.(arXiv:0304277)
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