Contents

Idea

Opetopes are one of the geometric shapes of cells in the approach to the higher category theory of n-categories and ω-categories.

References

An overview is in chapter 4 of

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

Opetopes were introduced here:

• John Baez and James Dolan, Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145–206. (arXiv)

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’:

• C. Hermida, Michael Makkai, and J. Power: On weak higher-dimensional categories I, II. Jour. Pure Appl. Alg. 157 (2001), 221–277.

• Michael Makkai, The multitopic $\omega$-category of all multitopic $\omega$-categories. (online)

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)