nLab
(infinity,2)-category

Idea

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

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

Models include

• the definition by Carlos Simpson and Tamsamani;
• the definition in terms of n-fold Segal spaces;
• a definition in terms of scaled simplicial sets, following Verity’s simplicial model for weak omega-categories by Jacob Lurie (see reference below)

See also the list of all definitions of higher categories at n-category.

Models for the $(\mathrm{\infty }\mathrm{,1})$-category of $(\mathrm{\infty }\mathrm{,2})$-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 $(\mathrm{\infty }\mathrm{,2})$-categories, in generalization of the standard model category models for (∞,1)-categories themselves (see there for details).

Recall that

• the standard model structure on simplicial sets models ∞-groupoids.

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

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

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

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

between Joyal-$\mathrm{SSet}$-enriched categories, Joyal-$\mathrm{SSet}$-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.

References

• Jacob Lurie, (∞,2)-Categories and the Goodwillie Calculus