nLab Homotopical Algebra



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

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

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

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\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



Paths and cylinders

Homotopy groups

Basic facts


This page is about the book

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


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. sAs A as a model category

5. Homology and cohomology

6. Modules over a simplicial ring

category: reference

Last revised on September 12, 2020 at 10:01:40. See the history of this page for a list of all contributions to it.