- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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).

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

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]

Last revised on February 7, 2024 at 07:33:03. See the history of this page for a list of all contributions to it.