---
created: 2023-03-31T12:02
updated: 2023-03-31T12:03
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---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 20:24:45
title: operad
aliases: [operad]

---


---

- Tags 
	- #homotopy/stable-homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [koszul dual](Unsorted/koszul%20duality.md)

---

# operad

Definition: a [multicategory](multicategory) with one object. 
Idea: generalized associative algebras.

![](attachments/Pasted%20image%2020220409234910.png)

![](2023-03-31-6.png)
![](2023-03-31-7.png)

# Associative operad
![](attachments/Pasted%20image%2020220317202245.png)

![](attachments/Pasted%20image%2020220316202504.png)
![](attachments/Pasted%20image%2020220316202552.png)

# Notes

![](attachments/Pasted%20image%2020210511200311.png)

Primary examples:
![](attachments/Pasted%20image%2020210511200339.png)

Given an operad $\OO$, the functor taking things to free $\OO\dash$algebras is a [monad](monad.md).

![](attachments/Pasted%20image%2020220320033531.png)
![](attachments/Pasted%20image%2020220320033604.png)
![](attachments/Pasted%20image%2020220320033635.png)

## In infty categories

![](attachments/Pasted%20image%2020220320033711.png)
Example: any [symmetric monoidal infty category](Unsorted/symmetric%20monoidal%20infty%20category.md), or the nerve of the category of operators on a classical colored operad.

Relation to [monad](Unsorted/monad.md):
![](attachments/Pasted%20image%2020220320034604.png)