nLab infinity-category



Higher category theory

higher category theory

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The term \infty-category broadly refers to higher categories with no bound on the dimension nn of their n 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=n = \infty, hence ( , 1 ) (\infty,1) -categories, have by now a very well developed theory with ubiquitous applications, and many authors refer to these ( , 1 ) (\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 ) (n,r) -categories, the previous general notion of \infty-categories may be referred to as (,)(\infty,\infty)-categories


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

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

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

in terms of inductively and coinductively defined equivalences:

and generalization to higher sheaves/stacks of these:

see also

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

Last revised on March 12, 2024 at 09:35:02. See the history of this page for a list of all contributions to it.