nLab (infinity,2)-category



Higher category theory

higher category theory

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Extra properties and structure

1-categorical presentations



An (,2)(\infty,2)-category is the special case of (,n)(\infty,n)-category for n=2n=2.

It is best known now through a geometric definition of higher category.

Models include:

See also the list of all definitions of higher categories at (∞,n)-category.


Models for the (,1)(\infty,1)-category of (,2)(\infty,2)-categories

In (∞,2)-Categories and the Goodwillie Calculus Jacob Lurie discusses a variety of model category structures, all Quillen equivalent, that all model the (∞,2)-category of (,1)(\infty,1)-categories, in generalization of the standard model category models for (∞,1)-categories themselves (see there for details).

Recall that

A simplicially enriched model category with respect to the standard model structure on simplicial sets hence models ∞Grpd-enriched categories, hence (∞,1)-categories.

Along this pattern (,2)(\infty,2)-categories should be modeled by categories enriched in the Joyal model structure that models the (∞,1)-category of (∞,1)-categories.

Write SSet JSSet^J for SSet equipped with the Joyal model structure. Then, indeed, there is a diagram of Quillen equivalences of model category structures

SSet JCatSSetSegSp[Δ op,SSet J] SSet^J Cat \to SSet SegSp \to [\Delta^{op}, SSet^J]

between Joyal-SSetSSet-enriched categories, Joyal-SSetSSet-enriched complete Segal spaces and simplicial Joyal-simplicial sets.

This is remark 0.0.4, page 5 of the article. There are many more models. See there for more.


Classes of examples include


Last revised on September 21, 2020 at 15:52:54. See the history of this page for a list of all contributions to it.