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:
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:
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
Eric Finster, Opetopic Diagrams 1 - Basics (video)
Eric Finster, Opetopic Diagrams 2 - Geometry (video)
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
Something like an implementation of aspects of opetopic type theory within homotopy type theory is described in
Last revised on December 16, 2018 at 00:59:23. See the history of this page for a list of all contributions to it.