nLab stable model category

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 $\mathcal{C}$ is

• a pointed model category

• such that the loop space object functor $\Omega$ and the suspension object functor $\Sigma$, are inverse equivalences on the homotopy category $Ho(C)$:

  $$\Omega \colon Ho(\mathcal{C}) \stackrel{\overset{\simeq}{\longleftarrow}}{\underset{\simeq}{\longrightarrow}} Ho(\mathcal{C}) \colon \Sigma \,.$$

Properties

Triangulated homotopy category

The homotopy category $Ho(\mathcal{C})$ of a stable model category, equipped with the reduced suspension functor $\Sigma \colon Ho(\mathcal{C})\overset{\simeq}{\to} Ho(\mathcal{C})$ 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 $A_\infty$-algebroid module categories

Proposition

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

• a simplicial model category;

• a cofibrantly generated model category;

• a proper model category;

• with a set $S$ of compact generators;

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

$$\mathcal{C} \simeq_Q A_S Mod \coloneqq Sp Cat(A_S, Sp)$$

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)

Remark

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:

Corollary

If if in prop. 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:

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

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 $A$ 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).

Related concepts

• linear model category

References

The concept originates with

• Mark Hovey, section 7 of Model Categories Mathematical Surveys and Monographs, Volume 63, AMS (1999) (pdf, Google books)

The classification theorems are due to

• Stefan Schwede, Brooke Shipley, Classification of stable model categories or Stable model categories are categories of modules, Topology 42 (2003) 103-153 (arXiv:0108143)

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)

On (monoidal) Bousfield localization of stable model categories:

• Luca Pol, Jordan Williamson, The Left Localization Principle, completions, and cofree $G$-spectra, J. Pure Appl. Algebra 224 11 (2020) 106408 [arXiv:1910.01410, doi:10.1016/j.jpaa.2020.106408]