---
created: 2021-11-04T01:28
updated: 2023-04-14T16:29
title: derived stacks
aliases: [derived stack, derived loop stack, derived moduli problem, moduli problem]

---

---

- Tags: 
	- #higher-algebra/derived 
- Refs:
	- #todo/add-references
- Links:
	- [stacks MOC](Unsorted/stacks%20MOC.md)
	- [[quasi-smooth]]
	- [Picard stack](Unsorted/Picard%20group.md)

---

# derived stacks

![](2023-04-14-2.png)
![](2023-04-14-3.png)

Motivation:
![](attachments/Pasted%20image%2020220516181716.png)
![](attachments/Pasted%20image%2020220516181736.png)

Conventions:
![](attachments/Pasted%20image%2020220516182300.png)

Derived extensions:
![](attachments/Pasted%20image%2020220516182347.png)

# Definitions

![](attachments/Pasted%20image%2020220903173318.png)

# Examples
The **derived loop stack**:

![Pasted image 20211104012824.png](Pasted%20image%2020211104012824.png)
![Pasted image 20211105134850.png](Pasted%20image%2020211105134850.png)

# Historical #why-care 

![Pasted image 20211105111739.png](Pasted%20image%2020211105111739.png)
![Pasted image 20211105111820.png](Pasted%20image%2020211105111820.png)
![Pasted image 20211105111919.png](Pasted%20image%2020211105111919.png)
![Pasted image 20211105111953.png](Pasted%20image%2020211105111953.png)

All moduli problems come with natural [dg Lie algebras](dg%20Lie%20algebras.md) describing their [formal completions](formal%20completion):

![Pasted image 20211105112122.png](Pasted%20image%2020211105112122.png)
![Pasted image 20211105132707.png](Pasted%20image%2020211105132707.png)

# Algebraic de Rham

![Pasted image 20211105132816.png](Pasted%20image%2020211105132816.png)
![Pasted image 20211105132917.png](Pasted%20image%2020211105132917.png)

Its [Hodge filtration](Unsorted/Hodge%20filtration.md)
![Pasted image 20211105132952.png](Pasted%20image%2020211105132952.png)

Closed forms are closer to [HH](Unsorted/HH.md)
![Pasted image 20211105133053.png](Pasted%20image%2020211105133053.png)

Symplectic structures:
![Pasted image 20211105133113.png](Pasted%20image%2020211105133113.png)
![Pasted image 20211105133120.png](Pasted%20image%2020211105133120.png)

Canonical example of $\GL_n$:
![Pasted image 20211105133218.png](Pasted%20image%2020211105133218.png)

![Pasted image 20211105133412.png](Pasted%20image%2020211105133412.png)

## Deformation Theory

![Pasted image 20211105134630.png](Pasted%20image%2020211105134630.png)
![Pasted image 20211105134819.png](Pasted%20image%2020211105134819.png)