It is well known that left [Bousfield localizations](http://ncatlab.org/nlab/show/Bousfield+localization) of the [global functor model category](http://ncatlab.org/nlab/show/global+model+structure+on+functors) $Func(C^{op}, SSet\_{standard})$ of functors with values in simplicial sets equipped with the [standard model structure on simplicial sets](http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets) yields the local [model structure on simplicial presheaves](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves) that models [hypercomplete oo-stacks](http://ncatlab.org/nlab/show/hypercomplete+(infinity%2C1)-topos).


What is known, though, about left Bousfield localizations of $Func(C^{op}, SSet\_{Joyal})$, with simplicial sets equipped with the [Joyal model structure](http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets#joyals_model_structure_9)?


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



Local Joyal-simplicial presheaves?
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](http://www.intlpress.com/HHA/v12/n2/a9/v12n2a9.pdf) of Clark Barwick (see also section 4.4 of Joseph Ayoub's [book](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF) 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](http://arxiv.org/abs/0901.3602) 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](http://arxiv.org/abs/math/9807049) 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](http://www.math.univ-toulouse.fr/~toen/hagI.pdf) and [Lurie's book](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf)), and obtain stacks of $(\infty,n)$-categories over any $(\infty,1)$-topos.


