nLab stable model category



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories




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 𝒞\mathcal{C} is


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


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)


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.



If if in prop. 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)


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 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).


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)

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.