#
nLab
Model Categories

### Context

#### Model category theory

**model category**, model $\infty$-category

**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*

There is the textbook

on model categories and the homotopy theory modeled by them.

## Preface

## Chapter 1. Model categories

### 1.1. The deﬁnition of a model category

### 1.2. The homotopy category

### 1.3. Quillen functors and derived functors

#### 1.3.1. Quillen functors

#### 1.3.2. Derived functors and naturality

#### 1.3.3. Quillen equivalences

### 1.4. 2-categories and pseudo-2-functors

## Chapter 2. Examples

### 2.1. Coﬁbrantly generated model categories

#### 2.1.1. Ordinals, cardinals, and transﬁnite compositions

#### 2.1.2. Relative I-cell complexes and the small object argument

#### 2.1.3. Coﬁbrantly generated model categories

### 2.2. The stable category of modules

### 2.3. Chain complexes of modules over a ring

### 2.4. Topological spaces

### 2.5. Chain complexes of comodules over a Hopf algebra

#### 2.5.1. The category of B-comodules

#### 2.5.2. Weak equivalences

#### 2.5.3. The model structure

## Chapter 3. Simplicial sets

### 3.1. Simplicial sets

### 3.2. The model structure on simplicial sets

### 3.3. Anodyne extensions

### 3.4. Homotopy groups

### 3.5. Minimal ﬁbrations

### 3.6. Fibrations and geometric realization

## Chapter 4. Monoidal model categories

### 4.1. Closed monoidal categories and closed modules

### 4.2. Monoidal model categories and modules over them

### 4.3. The homotopy category of a monoidal model category

## Chapter 5. Framings

### 5.1. Diagram categories

### 5.2. Diagrams over Reedy categories and framings

### 5.3. A lemma about bisimplicial sets

### 5.4. Function complexes

### 5.5. Associativity

### 5.6. Naturality

### 5.7. Framings on pointed model categories

## Chapter 6. Pointed model categories

### 6.1. The suspension and loop functors

### 6.2. Coﬁber and ﬁber sequences

### 6.3. Properties of coﬁber and ﬁber sequences

### 6.4. Naturality of coﬁber sequences

### 6.5. Pre-triangulated categories

### 6.6. Pointed monoidal model categories

## Chapter 7. Stable model categories and triangulated categories

### 7.1. Triangulated categories

### 7.2. Stable homotopy categories

### 7.3. Weak generators

### 7.4. Finitely generated model categories

## Chapter 8. Vistas

## Bibliography

## Index

Last revised on April 18, 2023 at 16:33:58.
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