---
date: 2022-03-18 10:29
modification date: Friday 18th March 2022 10:29:29
title: stable infinity category
aliases: [stable infinity category, stable infty category, loop, suspend, suspension]

---


---

- Tags 
	- #higher-algebra #homotopy/stable-homotopy 
- Refs:
	- <https://people.maths.ox.ac.uk/brantner/L6.pdf>
	- <https://www.math.purdue.edu/~pvankoug/talks/stableinfty16.pdf>
- Links:
	- [spectra](Unsorted/spectra.md)
	- [Ind construction](Ind%20construction.md)
	- [symmetric monoidal infty category](symmetric%20monoidal%20infty%20category.md)
	- [spaces](Unsorted/homotopy%20type.md)
	- [property vs structure](property%20vs%20structure.md)
	- [presentable](Unsorted/presentable%20category.md)
	- [accessible functor](accessible%20functor.md)

---

# Stable infty category


- Suspension and loops: $$\Suspend X \da 0 \glue{X} 0 \approx X[1] \qquad \Loops X \da 0\fiberprod{X} 0\approx X[-1]$$

![](attachments/Pasted%20image%2020220510160647.png)

# Suspension, Loops, fiber/cofiber sequences

![](attachments/Pasted%20image%2020220408154615.png)
![](attachments/Pasted%20image%2020220408154630.png)

![](attachments/Pasted%20image%2020220422153829.png)
![](attachments/Pasted%20image%2020220422153843.png)
![](attachments/Pasted%20image%2020220422153952.png)


# stable infinity category

![](attachments/Pasted%20image%2020220422153854.png)
![](attachments/Pasted%20image%2020220320032823.png)
![](attachments/Pasted%20image%2020220320032856.png)

![](attachments/Pasted%20image%2020220408192237.png)
![](attachments/Pasted%20image%2020220318102933.png)
![](attachments/Pasted%20image%2020220318103006.png)

![](attachments/Pasted%20image%2020220319001344.png)
![](attachments/Pasted%20image%2020220319001353.png)
![](attachments/Pasted%20image%2020220422153935.png)

## Exact triangles

![](attachments/Pasted%20image%2020220320032845.png)
![](attachments/Pasted%20image%2020220510234820.png)

# Suspension

The [loop](Unsorted/stable%20infinity%20category.md) and [suspension](Unsorted/stable%20infinity%20category.md) adjunction:

![](attachments/Pasted%20image%2020220320032951.png)

![](attachments/Pasted%20image%2020220510235021.png)
![](attachments/Pasted%20image%2020220510235033.png)
![](attachments/Pasted%20image%2020220510235043.png)
![](attachments/Pasted%20image%2020220510235347.png)

# Fibers

See [arrow category](arrow%20category).
![](attachments/Pasted%20image%2020220320033021.png)

# Fiber sequences
![](attachments/Pasted%20image%2020220510235437.png)
![](attachments/Pasted%20image%2020220510235459.png)

# Grothendieck group and G-theory

See [K-theory](Unsorted/K-theory.md) and [G-theory](G-theory):
![](attachments/Pasted%20image%2020220320033230.png)

# Connectivity

![](attachments/Pasted%20image%2020220320033309.png)

# Spectra

![](attachments/Pasted%20image%2020220422154616.png)
![](attachments/Pasted%20image%2020220422154119.png)
![](attachments/Pasted%20image%2020220422154129.png)
![](attachments/Pasted%20image%2020220510235541.png)
![](attachments/Pasted%20image%2020220510235636.png)
![](attachments/Pasted%20image%2020220510235800.png)
![](attachments/Pasted%20image%2020220510235855.png)
![](attachments/Pasted%20image%2020220510235938.png)
![](attachments/Pasted%20image%2020220511000144.png)

## Adjunction

See [spectrum objects in an infinity category](spectrum%20objects%20in%20an%20infinity%20category.md)
![](attachments/Pasted%20image%2020220422154651.png)
![](attachments/Pasted%20image%2020220422154826.png)

![](attachments/Pasted%20image%2020220422155356.png)

# Exactness

![](attachments/Pasted%20image%2020220422154739.png)

# Duality

![](attachments/Pasted%20image%2020220422155506.png)
![](attachments/Pasted%20image%2020220422155524.png)
![](attachments/Pasted%20image%2020220422155556.png)