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
and
nonabelian homological algebra
A stable model category is a 1-category structure used to present a stable (∞,1)-category in analogy to how a general model category encodes a general (∞,1)-category.
A stable model category $C$ is
such that the loop space object functor $\Omega$ and the suspension object functor $\Sigma$, are inverse equivalences on the homotopy category $Ho(C)$:
Let $C$ be a stable model category that is in addition
then there is a chain of sSet-enriched Quillen equivalences linking $C$ to the the spectrum-enriched functor category
equipped with the global model structure on functors, where $\mathcal{E}(S)$ is the $Sp$-enriched category given by
This is theorem 3.3.3 in (Schwede-Shipley)
Notice the similarity (but superficial difference: $sSet$/$Sp$-enrichment localization/no-localization) to the stable Giraud theorem discussed at stable (∞,1)-category.
Moreover, by Schwede-Shipley 03 theorems, 3.1.1, 3.3.3, 3.8.2 stable model categories equivalent (by zig-zags of Quillen equivalences) to categories of module spectra over some ring spectrum. If that is an Eilenberg-MacLane spectrum, then this identifies the corresponding stable model categories with the model structure on unbounded chain complexes.
Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).
The standard accounts are
Discussion of the notion of stable model categories with the abstract notion of stabilization in (infinity,1)-category theory is in section 4.2 (prop. 4.15) of