---
created: 2021-10-11T17:52
updated: 2023-04-14T17:08
title: E_n ring spectrum
aliases: ["E_n ring spectrum", "E_n algebra", "E_n ring", "E_infty ring", "E_infty algebra", "E_infty", E-infty ring spectrum, E-infty ring, module spectra, module spectrum, ring spectra, ring spectrum, Gamma space]
---

---

- Tags 
	- #homotopy/stable-homotopy 
- Refs:
	- #todo/add-references
- Links:
	- [spectra](Unsorted/spectra.md)
	- [symmetric monoidal](monoidal%20category.md)

---

# $\EE_\infty$ ring spectra

![](2023-04-14.png)
![](2023-04-14-1.png)


![](attachments/Pasted%20image%2020220511000543.png)
![](attachments/Pasted%20image%2020220511001217.png)

More or less the same thing as a $\Gamma\dash$space a la Segal.
Can produce these from an [H-space](Unsorted/H-space.md)


![](attachments/Pasted%20image%2020220318104013.png)
![](attachments/Pasted%20image%2020220318110308.png)

# $\EE_n$ ring spectra

Relation to [surgery](Unsorted/surgery.md):

![Pasted image 20210717012125](attachments/Pasted%20image%2020210717012125.png)


![](attachments/Pasted%20image%2020220209185005.png)

## Modules over a ring spectrum

![](attachments/Pasted%20image%2020220209185529.png)

# Notes

Definitions of [tor amplitude](Unsorted/tor%20amplitude.md), [perfect complexes](Unsorted/perfect%20complexes.md), and [coherent modules](Unsorted/coherent%20module.md):
![](attachments/Pasted%20image%2020220209185708.png)

Relation to [structured ring spectra](structured%20ring%20spectrum) and [symmetric spectra](Unsorted/symmetric%20spectra.md)
![](attachments/Pasted%20image%2020220316202343.png)

# Examples

![](attachments/Pasted%20image%2020220209185317.png)


# E_n algebra objects in categories

![](2023-04-14-13.png)

We can summarize Definition 6.1 informally as follows: an $E_{n^{-}}$ algebra object of a symmetric monoidal $\infty$-category $\mathcal{C}$ is an object $A \in \mathcal{C}$ which is equipped with $n$ multiplication operations $\left\{m_i: A \otimes A \rightarrow A\right\}_{1 \leq i \leq n}$;
these multiplications are required to be associative and unital (up to coherent homotopy) and to be compatible with one another in a suitable sense.

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