---
date: 2021-11-05 11:16
modification date: Friday 5th November 2021 11:16:03
title: monoidal category
aliases: [Monoidal category, monoidal categories, symmetric monoidal category, symmetric monoidal, cosmos, braided monoidal, braided]

---


---

- Tags 
	- #higher-algebra 
- Refs:
	- #todo/add-references
- Links:
	- [Drinfeld center](Unsorted/Drinfeld%20center.md)
	- [Calabi-Yau object of a category](Unsorted/Calabi-Yau%20object%20of%20a%20category.md)
	- [linear category](Unsorted/linear%20category.md)

---

# monoidal category

![Pasted image 20211105111610.png](Pasted%20image%2020211105111610.png)
![Pasted image 20211105111621.png](Pasted%20image%2020211105111621.png)


![](attachments/Pasted%20image%2020220316210159.png)

![](attachments/Pasted%20image%2020220316210213.png)

# Cosmos
A symmetric closed-monoidal category that admits small limits and small colimits.

# In infty cats

![](attachments/Pasted%20image%2020220510160509.png)

![](attachments/Pasted%20image%2020220318003549.png)


# Dualizability and invertibility

![](attachments/Pasted%20image%2020220510160551.png)

A category is [rigid](rigid) iff all objects are monoidally invertible. 

# For infty categories

![](attachments/Pasted%20image%2020220510233830.png)
![](attachments/Pasted%20image%2020220510233850.png)