# nLab stable model category

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

#### Stable homotopy theory

stable homotopy theory

Introduction

# 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).

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)