---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 20:11:03
title: Model category theory
aliases: [Model category theory, "model structure", "model category", model cat, model]

---


---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [stable infinity category](Unsorted/stable%20infinity%20category.md)

---

# Model category theory

![](attachments/Pasted%20image%2020220319001253.png)

# Definitions

- Definition of a model category:
![](attachments/Pasted%20image%2020210505015517.png)

![](attachments/Pasted%20image%2020210516135205.png)


# Topics

[classical category theory](Unsorted/classical%20category%20theory.md)
- [Serre fibration](Serre fibration.md)
- [Fibrant object](Fibrant object)
- [Cofibrant object](Cofibrant object)
- [Cofibrant generation](Cofibrant generati
- [Weak equivalence](Weak equivalence.md)
- [localization (category theory)](localization (category theory).md)
- [the model structure on spectra](the model structure on spectra)

## Applies to Chain Complexes
- [derived functor](derived functor.md)
- [homotopy category](homotopy category.md)
- [The model structure on chain complexes](The model structure on chain complexes)

## Model Cat Specific
- [Quillen equivalence](Quillen equivalence.md)
- [Quillen adjunction](Quillen adjunction.md)
-on)
- [The model structure on simplicial sets](The model structure on simplicial sets.md)

# Notes
Let $\nerve{\wait}_{\dg}$ be the DG [nerve](nerve.md) functor, then

![](attachments/Pasted%20image%2020210516135955.png)

Being local:
![](attachments/Pasted%20image%2020210603191103.png)
![](attachments/Pasted%20image%2020210603191120.png)

- Important: the [small object argument](small object argument)


![](attachments/Pasted%20image%2020220316200147.png)
![](attachments/Pasted%20image%2020220316200157.png)
![](attachments/Pasted%20image%2020220316200213.png)
![](attachments/Pasted%20image%2020220316200241.png)
![](attachments/Pasted%20image%2020220316200256.png)

![](attachments/Pasted%20image%2020220316200325.png)
![](attachments/Pasted%20image%2020220316200401.png)

See [Quillen adjunction](Unsorted/Quillen%20adjunction.md)

## Model structure on sSet

![](attachments/Pasted%20image%2020220316200743.png)

## Model structure on Top

![](attachments/Pasted%20image%2020220316200835.png)

## Model structure on spectra

![](attachments/Pasted%20image%2020220316201102.png)

# Tensored

![](attachments/Pasted%20image%2020211003194259.png)