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stable model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Stable homotopy theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

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.

Defintion

A stable model category 𝒞\mathcal{C} is

Properties

Triangulated homotopy category

The homotopy category Ho(𝒞)Ho(\mathcal{C}) of a stable model category, equipped with the reduced suspension functor Σ:Ho(𝒞)Ho(𝒞)\Sigma \colon Ho(\mathcal{C})\overset{\simeq}{\to} Ho(\mathcal{C}) is a triangulated category (Hovey 99, section 7).

Relation to stable \infty-categories

Stabilization of model categories is a model for the abstractly defined stabilization in (infinity,1)-category theory (Robalo 12, prop. 4.15).

As A A_\infty-algebroid module categories

Proposition

Let 𝒞\mathcal{C} be a stable model category that is in addition

then there is a chain of sSet-enriched Quillen equivalences linking 𝒞\mathcal{C} to the the spectrum-enriched functor category

𝒞 QA SModSpCat(A S,Sp) \mathcal{C} \simeq_Q A_S Mod \coloneqq Sp Cat(A_S, Sp)

equipped with the global model structure on functors, where A SA_S is the SpSp-enriched category whose set of objects is SS

This is in (Schwede-Shipley, theorem 3.3.3)

Remark

An SpSp-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 SpSp-enriched category is an A A_\infty-ringoid. It is has a single object then (as a pointed category) it is an A-infinity algebra.

Hence:

Corollary

If if in prop. 1 there is just one compact generator P𝒞P \in \mathcal{C}, then there is a one-object SpSp-enriched category, hence an A-infinity algebra AA, which is the endomorphisms AEnd 𝒞(P)A \simeq End_{\mathcal{C}}(P), and the stable model category is its category of modules:

𝒞 QAMod. \mathcal{C} \simeq_Q A Mod \,.

This is in (Schwede-Shipley, theorem 3.1.1)

Remark

This may be thought of as a homotopy-theoretic analog of the Freyd-Mitchell embedding theorem for abelian categories.

Remark

One way to read this is that formal duals of presentable stable infinity-categories are a model for spaces in (“derived”) noncommutative geometry.

If AA 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).

References

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)

Revised on April 20, 2016 15:54:14 by Urs Schreiber (131.220.184.222)