nLab opetope



Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



Opetopes are one of the geometric shapes of cells in the approach to the higher category theory of n-categories and ∞-categories put forward in (Baez-Dolan 97) and developed by (Makkai) and others: opetopic ∞-categories.

A syntactic formalization of opetopic ∞-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’:

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

  • Eugenia Cheng, Weak nn-categories: opetopic and multitopic foundations, Jour. Pure Appl. Alg. 186 (2004), 109–137.(arXiv)

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


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

  • Eric Finster, Type theory and the opetopes, talk at HDACT Ljubljana, June 2012 (pdf)

Something like an implementation of aspects of opetopic type theory within homotopy type theory is described in

Last revised on April 3, 2020 at 08:11:03. See the history of this page for a list of all contributions to it.