nLab weak omega-category

Idea

In higher category theory the term weak $\omega$-categories is essentially synonymous with infinity-category in the fully general sense of (infinity,infinity)-category. The terms “$\omega$-category” and “$\infty$-category” originate in different schools and their choice of use is mostly a matter of the preference of individual authors.

One slight difference is that “$\infty$-category” usually implies a “weak” (fully general) notion, while in addition to weak $\omega$-categories there are also strict ones. Another difference is that definitions of weak $\omega$-categories tend to be algebraic instead of geometric (accordingly typically the central open question is whether a definition really satisfies the homotopy hypothesis), though some definitions of weak $\omega$-categories are geometry (for instance some flavors of definition of opetopic omega-category).

Examples

The following are examples for proposals of definitions of weak $\omega$-categories.

• Trimble ∞-category

• simplicial weak ∞-category

• Batanin ∞-category

• opetopic omega-category

References

Fore more see general references at higher category theory, such as:

• Andre Joyal, Tim Porter, Peter May, Weak categories (pdf)

Discussion of weak $\omega$-categories via computads construed as inductive types:

• Christopher J. Dean, Eric Finster, Ioannis Markakis, David Reutter, Jamie Vicary, Computads for weak $\omega$-categories as an inductive type [arXiv:2208.08719]