---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 19:57:47
title: monad
aliases: [monads, algebra over a monad]

---


---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Barr-Beck](Unsorted/Barr-Beck.md)

---

# monad

![](attachments/Pasted%20image%2020220318100858.png)
![](attachments/Pasted%20image%2020220318101112.png)
![](attachments/Pasted%20image%2020220511004425.png)



![](attachments/Pasted%20image%2020220316203158.png)

![](attachments/Pasted%20image%2020220207150139.png)

# Algebras over monads

![](attachments/Pasted%20image%2020220317235437.png)

![](attachments/Pasted%20image%2020220511004528.png)
![](attachments/Pasted%20image%2020220511004539.png)

# Free resolutions

![](attachments/Pasted%20image%2020220318000350.png)

# Examples

To produce resolutions: let $Q\da f_* f_*$ and $(X, \OO_X)$ be a ringed space and $\mcf \in \mods{\OO_X}$, there is a resolution $0\to \mcf \to Q\mcf \to QQ\mcf \to \cdots$, and on stalks there is a homotopy equivalence $\mcf[0] \sim (\mcf_x \to (Q\mcf)_x \to \cdots) \in \Ch(\mods{\OO_{X, x}})$.

![](attachments/Pasted%20image%2020220317235412.png)
![](attachments/Pasted%20image%2020220317235421.png)![](attachments/Pasted%20image%2020220317235508.png)

# Adjunctions

![](attachments/Pasted%20image%2020220318000024.png)


# Recognition theorem

See [Morita equivalence](Unsorted/Morita%20equivalence.md)

![](attachments/Pasted%20image%2020220318110629.png)
![](attachments/Pasted%20image%2020220318110639.png)
![](attachments/Pasted%20image%2020220318110830.png)