nLab Model Categories


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

There is the textbook

on model categories and the homotopy theory modeled by them.


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



category: reference

Last revised on April 18, 2023 at 16:33:58. See the history of this page for a list of all contributions to it.