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)

Last revised on July 6, 2015 at 11:12:08. See the history of this page for a list of all contributions to it.