nLab opetope

Idea

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).

Remark

(etymology) Judging from the abstract of Baez & Dolan 1997, the word “opetope” seems to derive from operad/operation + polytope. The paper notes that the first two syllables are meant to be pronounced as in “operation”.

References

An overview is in chapter 4 of

• Eugenia Cheng, Aaron Lauda, Higher dimensional categories: an illustrated guidebook (pdf)

and in chapter 7 of

• Tom Leinster, Higher operads, higher categories, London Math. Soc. Lec. Note Series 298, math.CT/0305049

Opetopes were introduced here:

• John Baez, James Dolan, Higher-dimensional algebra 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)

• 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:

• Eugenia Cheng, Opetopic bicategories: comparison with the classical theory. (arXiv)

A higher dimensional string diagram-notation for opetopes was introduced (as “zoom complexes” in section 1.1) in

• Joachim Kock, André Joyal, Michael Batanin, Jean-François Mascari, Polynomial functors and opetopes (arXiv:0706.1033)

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

• Thorsten Palm, …

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

• Eric Finster, Towards Higher Universal Algebra in Type Theory, Homotopy Type Theory Electronic Seminar 2018 (recording, Agda code)