# Local Joyal-simplicial presheaves?

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model structure on simplicial sets yields the local model structure on simplicial presheaves that models hypercomplete oo-stacks.

What is known, though, about left Bousfield localizations of $Func(C^{op}, SSet_{Joyal})$, with simplicial sets equipped with the Joyal model structure?

If it exists, this should model $(\infty,2)$-sheaves on $C$. But possibly it doesn't exist.

-

If $V$ is a reasonnable model category (i.e. combinatorial, etc), and $C$ a small category endowed with a Grothendieck topology $\tau$, there are $\tau$-local model structures on the category $Fun(C^{op},V)$ (a projective version as well as an injective version). You may have a look at this paper of Clark Barwick (see also section 4.4 of Joseph Ayoub's book for the hypercomplete versions). In the case $V$ is the standard model structure on simplicial sets, we get the usual homotopy theory of stacks in $\infty$-groupoids, while, if $V$ is the Joyal model structure, you get the homotopy theory of stacks in $(\infty,1)$-categories. If you consider the hypercomplete version, then these model structures will behave in the usual way; for instance, if moreover the topos of sheaves on $C$ has enough points, a morphism of simplicial presheaves on $C$ will be a weak equivalence for the hypercomplete $\tau$-local Joyal model structure if and only if its stalkwise is a weak equivalence for the Joyal model structure.
You may as well consider the case where $V$ is the Rezk model structure for $(\infty,n)$-categories to get the homotopy theory of stacks in $(\infty,n)$-categories (for $0\leq n\leq \infty$) to obtain the $(\infty,n+1)$-topos of stacks in $(\infty,n)$-categories (whatever this means). Alternatively, you may as well consider the case where $V$ is the model category of Segal $n$-categories, and get something rather close to Simpson and Hirschowitz theory of higher stacks. It is also possible to consider the case where $C$ is a simplicial category and $\tau$ is a Grothendieck topology on $C$ (in the sense of Toën-Vezzosi-Lurie, see HAG I and Lurie's book), and obtain stacks of $(\infty,n)$-categories over any $(\infty,1)$-topos.