Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
Stable homotopy theory
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 (generally non-stable) (∞,1)-category.
A stable model category is
Triangulated homotopy category
The homotopy category of a stable model category, equipped with the reduced suspension functor is a triangulated category (Hovey 99, section 7).
Relation to stable -categories
Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).
As -algebroid module categories
This is in (Schwede-Shipley, theorem 3.3.3)
If if in prop. 1 there is just one compact generator , then there is a one-object -enriched category, hence an A-infinity algebra , which is the endomorphisms , and the stable model category is its category of modules:
This is in (Schwede-Shipley, theorem 3.1.1)
If 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 concept originates with
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
- Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)