(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
The stabilization of an (∞,1)-topos $\mathbf{H}$
consist of spectrum objects in $\mathbf{H}$. By the “stable Giraud theorem” this is the localization of an (∞,1)-category of (∞,1)-functors with values in the stable (∞,1)-category of spectra: $\infty$-sheaves of spectra.
This may be presented by a model structure on presheaves of spectra.
The stable Dold-Kan correspondence
is an (∞,1)-limits-preserving (∞,1)-functor from the (∞,1)-category of chain complexes (over a given commutative ring) to the (∞,1)-category of spectra. Hence every (∞,1)-sheaf/∞-stack of chain complexes (as it appears (maybe implicitly) in abelian sheaf cohomology/hypercohomology canonically incarnates as an $(\infty,1)$-sheaf of spectra).
The smash tensor product of the symmetric monoidal (∞,1)-category of spectra induced also a smash tensor product on any (∞,1)-category of sheaves of spectra
(Lurie, "Spectral Schemes", prop 1.5).
The homotopy categories of sheaves of combinatorial spectra were discussed in
A model category structure of presheaves of spectra akin to the model structure on simplicial presheaves is discussed in
Plenty of further discussion in terms of model category theory is in
Discussion in terms of (∞,1)-category/(∞,1)-topos-theory is in
Bertrand Toën, section 1.2 of K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems (arXiv:math/9908097)
Michael Paluch, Algebraic K-theory and topological spaces (pdf)