nLab opetopic omega-category



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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.


By Baez-Dolan

(Baez-Dolan 97)

By Makkai

(Hermida-Makkai-Power 01, Makkai)

The actual definition of opetopic ω\omega-categories, called “multitopic ω\omega-categories” here, appears on p.57 of (Makkai).

By Palm

In the definition by (Palm) of opetopic ω\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:

  1. 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.

  2. 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:

Some mistakes were corrected in subsequent papers:

Makkai and collaborators introduced a slight variation they called ‘multitopes’:

Cheng has carefully compared opetopes and multitopes, and various approaches to opetopic nn-categories:

  • Eugenia Cheng, Weak nn-categories: comparing opetopic foundations, Jour. Pure Appl. Alg. 186 (2004), 219–231.


  • Eugenia Cheng, Weak nn-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

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

Last revised on February 10, 2024 at 02:40:21. See the history of this page for a list of all contributions to it.