---
date: 2022-02-23 18:45
modification date: Wednesday 23rd March 2022 17:36:14
title: cofiber sequence
aliases: [cofiber, fiber, cone, cocone, kernel, cokernel, homotopy kernel, homotopy cokernel, hocofib, cofib, fib, hofib]

---

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


---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [spectra are modules over the sphere spectrum](spectra%20are%20modules%20over%20the%20sphere%20spectrum.md)

---

# cofiber sequence

- Homotopy fibers:
![](attachments/Pasted%20image%2020210505014535.png)
![](attachments/Pasted%20image%2020210505014555.png)


Relation to classical notions:

- $\mathrm{fib} \sim \mathrm{hoker} \sim \cone$ 
- $\mathrm{cofib} \sim \mathrm{hocoker} \sim\mathrm{cocone}$

Generally: $\fib(A\to B) = \hocolim(A\to B\from \pt)$ is a pushout and $\cofib(A\to B) = \hocolim(\pt \to B \from A)$ is a pullback.![](attachments/Pasted%20image%2020220401203156.png)

![](attachments/Pasted%20image%2020220401204136.png)
![](attachments/Pasted%20image%2020220401204203.png)
![](attachments/Pasted%20image%2020220401204306.png)

![](attachments/Pasted%20image%2020220422200053.png)