on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
on strict ∞-categories?
For $V$ a sufficiently nice (monoidal) model category and $C$ a small category equipped with a Grothendieck topology $\tau$, there are left Bousfield localizations of the global model structure on functors $[C^{op}, V]$ whose fibrant objects satisfy descent with respect to ?ech cover?s or even hypercovers with respect to $\tau$.
These model structures are expected to model $V$-valued ∞-stacks on $C$. This is well understood for the case $V =$ SSet equipped with the standard model structure on simplicial sets modelling ∞-groupoids. In this case the resulting local model structure on simplicial presheaves is known to be one of the models for ∞-stack (∞,1)-toposes.
But the general localization procedure works for choices of $V$ different from and more general than SSet with its standard model structure. In particular it should work for
$V =$ SSet equipped with its Joyal model structure, modelling (∞,1)-categories
$V = (n,r)\Theta Spaces$, the model structure on Theta spaces modelling weak (n,r)-categories;
$V = dSet$, the model structure on dendroidal sets modelling (∞,1)-operads.
For these cases the local model structure on $V$-valued presheaves should model, respectively, $(n,r)$-category valued sheaves/stacks and $(\infty,1)$-operad valued sheaves/stacks.
The general localization result is apparently due to
which considers the ?ech cover?-localization assuming $V$ to be monoidal and
which apparently does the hypercover descent and without assuming $V$ to be monoidal.
Much of this was kindly pointed out by Denis-Charles Cisinski in discussion here.