#
nLab

Homotopical Algebra

### 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

#### specific

### 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

#### Homotopy theory

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

This page is about the book

introducing the theory of model categories as a tool in homotopy theory.

# Contents

## Chapter I. Axiomatic homotopy theory

### 1. The axioms

### 2. The loop and suspension functors

### 3. Fibration and cofibration sequences

### 4. Equivalence of homotopy theories

### 5. Closed model categories

## Chapter II. Examples of simplicial homotopy theories

### 1. Simplicial categories

### 2. Closed simplicial model categories

### 3. Topological spaces, sumplicial sets, and simplicial groups

### 4. $s A$ as a model category

### 5. Homology and cohomology

### 6. Modules over a simplicial ring