nLab infinity-category

Idea

The term $\infty$-category broadly refers to higher categories with no bound on the dimension $n$ of their $n$-morphisms.

There are two different ways this is understood more in detail:

Historically, a tower of notions of n-categories (0-, 1-, 2-, 3-, 4-categories, …) was continued to a limiting case of “omega-categories”, and sometimes “$\infty$-categories” is used synonymously in this sense, broadly referring to the most general notion of higher categories. Beware that this general notion is not well developed nor is the equivalence or not of the few existing models, but see for instance the entries weak omega-category, weak complicial set and opetopic omega-categories .

The more fine-grained organization of types of higher categories by bidegree is better understood: (n,r)-categories for $n = \infty$, hence $(\infty,1)$-categories, have by now a very well developed theory with ubiquitous applications, and many authors refer to these $(\infty,1)$-categories (aka: quasi-categories) as just “$\infty$-categories”, for short (some authors even just say “categories” now, for this case, with a homotopy-theoretic background foundation tacitly understood).

In terms of the terminology of $(n,r)$-categories, the previous general notion of $\infty$-categories may be referred to as $(\infty,\infty)$-categories

Related entries

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (n,1)-category

• (∞,1)-category, internal (∞,1)-category, ∞-groupoid

  table - models for (∞,1)-categories

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

• weak omega-category

• opetopic omega-category

References

See also the references at (∞,1)-category, (∞,n)-category, (n,r)-category.

A versatile approach via enriched category theory and infinity-cosmoi:

• Emily Riehl, Dominic Verity, Infinity category theory from scratch, Higher Structures 4 1 (2020) [arXiv:1608.05314, pdf, lectures]

On (∞,1)-categories of (∞,∞)-categories

in terms of inductively and coinductively defined equivalences:

• Zach Goldthorpe, Homotopy theories of $(\infty,\infty)$-categories as universal fixed points with respect to enrichment, International Mathematics Research Notices 2023 22 (2023) 19592–19640 [arXiv:2307.00442, doi:10.1093/imrn/rnad196]

and generalization to higher sheaves/stacks of these:

• Zach Goldthorpe, Sheaves of $(\infty,\infty)$-categories [arXiv:2403.069262]

see also

• Félix Loubaton, Theory and models of $(\infty,\omega)$-categories [arXiv:2307.11931]