---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 20:43:50
title: Kan extension
aliases: [Kan extension]

---


---

- Tags 
	- #higher-algebra/infty-cats 
- Refs: #resources 
	- [Unsorted/Introduction to infinity categories](Unsorted/Introduction%20to%20infinity%20categories.md) 
	- [Kan complex](Kan%20complex.md)
	- Thesis: Everything is a Kan extension 
	  <https://www.math.harvard.edu/media/lehner.pdf> #resources/theses 
- Links:
	- [infinity category](Unsorted/infinity%20categories.md)

---

# Kan extension

# Motivation

Strictly generalizes a [colimit](Unsorted/limit%20(categorical).md):
![](attachments/Pasted%20image%2020220319191738.png)
![](attachments/Pasted%20image%2020220319191807.png)

# Definition

^4b6582

![](attachments/image_2021-03-25-00-08-55.png) ^492191


![](attachments/Pasted%20image%2020220316204349.png)![](attachments/Pasted%20image%2020220316204455.png)

![](attachments/Pasted%20image%2020220319191507.png)
![](attachments/Pasted%20image%2020220320040049.png)

# Example

![](attachments/Pasted%20image%2020220319191443.png)![](attachments/Pasted%20image%2020220325164118.png)
![](attachments/Pasted%20image%2020220325164137.png)