---
date: 2022-01-15 21:49
modification date: Sunday 13th February 2022 16:33:49
title: tensor category
aliases: [tensor ideal, tensor categories]

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Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #higher-algebra/monoidal 
- Refs:
	- <https://ocw.mit.edu/courses/18-769-topics-in-lie-theory-tensor-categories-spring-2009/pages/lecture-notes/>
- Links:
	- [triangulated category](Unsorted/triangulated%20category.md)
	- [compactly generated](Unsorted/compact%20object%20of%20a%20category.md)

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# tensor category

- Many local notions and constructions for $\rmod$ over some $R\in\CRing$ can be carried out in an arbitrary (linear) cocomplete tensor category.

Classification of thick tensor ideals and localizing tensor ideals as the key to capturing the global structure of tensor categories; construction of novel support theories.

Important: the [tensor-triangulated category](tensor-triangulated%20category).

![](attachments/Pasted%20image%2020220213163352.png)

# Tensor ideals
![](attachments/Pasted%20image%2020220213163411.png)

# Stable module categories

![](attachments/Pasted%20image%2020220213163431.png)

# Compact and compactly generated
See [compact](Unsorted/compact%20object%20of%20a%20category.md)
![](attachments/Pasted%20image%2020220213163449.png)