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 $Ho(C)$:
Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).
Let $\mathcal{C}$ be a stable model category that is in addition
then there is a chain of sSet-enriched Quillen equivalences linking $\mathcal{C}$ to the the spectrum-enriched functor category
equipped with the global model structure on functors, where $A_S$ is the $Sp$-enriched category whose set of objects is $S$
This is in (Schwede-Shipley, theorem 3.3.3)
An $Sp$-enriched category is a homotopy-theoretic analog of an Ab-enriched category, which may be thought of as a many-object version of a ring, a “ringoid”. Accordingly, an $Sp$-enriched category is an $A_\infty$-ringoid. It is has a single object then (as a pointed category) it is an A-infinity algebra.
Hence:
If if in prop. 1 there is just one compact generator $P \in \mathcal{C}$, then there is a one-object $Sp$-enriched category, hence an A-infinity algebra $A$, which is the endomorphisms $A \simeq End_{\mathcal{C}}(P)$, and the stable model category is its category of modules:
This is in (Schwede-Shipley, theorem 3.1.1)
This may be thought of as a homotopy-theoretic analog of the Freyd-Mitchell embedding theorem for abelian categories.
Hence one way to read this is to say that presentable stable infinity-categories are a model for (“derived”) noncommutative geometry.
If $A$ is an Eilenberg-MacLane spectrum, then this identifies the corresponding stable model categories with the model structure on unbounded chain complexes.
This is (Schwede-Shipley 03, theorem 5.1.6).
The classification theorems are due to
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