nLab stable model category

Contents

model category

Model structures

for ∞-groupoids

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

Homological algebra

homological algebra

Introduction

diagram chasing

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 $\mathcal{C}$ is

• 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

• 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).

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)

Last revised on April 20, 2016 at 15:54:14. See the history of this page for a list of all contributions to it.