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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.

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up vote 7 down vote accepted

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.

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Thanks a whole lot! But to apply general results on existence of Bousfield localization, don't we need that the homotopy classes of local weak equivalences forms a small set? For the local model structure on simplicial presheaves Joyal-Jardine's construction may be regarded as proving that the localization exists event though there is not a small set of local weak equivalences. Is the generalization of this actually in the literature? (I'll have a look at the Barwick reference, it seems I am aware of a similar but different document). –  Urs Schreiber Nov 17 '09 at 14:01
    
It is to be expected that the localization of (oo,n)-cat valued presheaves using the Rezk model structure yields the Simpson-Hirschowitz construction. But is this actually known in any detail? –  Urs Schreiber Nov 17 '09 at 14:02
    
For the existence of the left Bousfield localizations, the proof of Barwick and Ayoub consist indeed to show it is sufficient to invert a small set of maps (the difficulty is essentially the same as for the classical case). As for the comparisons between Rezk model structures and model structures for Segal higher categories, they are not written yet, unfortunately (even though it is true). –  Denis-Charles Cisinski Nov 17 '09 at 14:22
    
And thirdly, since I kinddly have your attention: what I am really interested in is playing this game with your model structore on dendroidal sets, i.e. consider local dendroidal set valued presheaves. Have you thought about that? –  Urs Schreiber Nov 17 '09 at 14:22
    
The case where V is the model structure on dendroidal sets works well too. What kind of problem do you have in mind? –  Denis-Charles Cisinski Nov 17 '09 at 14:33

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