model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
Theorems
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
such that the loop space object functor and the suspension object functor , are inverse equivalences on the homotopy category :
The homotopy category of a stable model category, equipped with the reduced suspension functor is a triangulated category (Hovey 99, section 7).
Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).
Let be a stable model category that is in addition
then there is a chain of sSet-enriched Quillen equivalences linking to the the spectrum-enriched functor category
equipped with the global model structure on functors, where is the -enriched category whose set of objects is
This is in (Schwede-Shipley, theorem 3.3.3)
An -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 -enriched category is an -ringoid. It is has a single object then (as a pointed category) it is an A-infinity algebra.
Hence:
If if in prop. 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)
This may be thought of as a homotopy-theoretic analog of the Freyd-Mitchell embedding theorem for abelian categories.
One way to read this is that formal duals of presentable stable infinity-categories are a model for spaces in (“derived”) noncommutative geometry.
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
On (monoidal) Bousfield localization of stable model categories:
Last revised on April 23, 2023 at 03:35:27. See the history of this page for a list of all contributions to it.