# nLab (infinity,2)-category

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

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

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.

## Properties

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

In (∞,2)-Categories and the Goodwillie Calculus Jacob Lurie discusses a variety of model category structures, all Quillen equivalent, that all model the (∞,1)-category of $(\infty,2)$-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 $(\infty,2)$-categories should be modeled by categories enriched in the Joyal model structure that models the (∞,1)-category of (∞,1)-categories.

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

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

between Joyal-$SSet$-enriched categories, Joyal-$SSet$-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.

## Examples

Classes of examples include

## References

Revised on August 1, 2014 12:57:22 by Colin Zwanziger (174.63.87.107)