# nLab linear 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

## Definition

A model category is called linear if it has a zero object (is a “pointed category” and hence a pointed model category) and for all of its objects $X$, the unit

$X \stackrel{\simeq}{\longrightarrow} \Omega \Sigma X$

of the (reduced suspension $\dashv$ loop space object)-adjunction is a weak equivalence.

## References

Last revised on January 31, 2021 at 08:08:23. See the history of this page for a list of all contributions to it.