nLab opetopic omega-category

Idea

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.

Definitions

By Baez-Dolan

(Baez-Dolan 97)

By Makkai

(Hermida-Makkai-Power 01, Makkai)

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

By Palm

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:

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.

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: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.

  (pdf, web)

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.

  (arXiv:0304279)

• 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:

• 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 Polish Seminar on Category Theory and its Applications, June 2012 (pdf)