---
date: 2021-04-26
tags: [ web/quick-notes ]
---


# 2021-04-26

## Random Notes

Some random notes: #todo

-Working out relative homology, an example: 
![](attachments/image_2021-04-25-01-52-05.png) 

- Chain of implications for module properties: 
![](attachments/image_2021-04-25-01-52-56.png)

- Definitions of common matrix groups: 
![](attachments/image_2021-04-25-01-53-18.png)

- Good example of exact triangles: 
![](attachments/image_2021-04-25-01-53-49.png)

- Manifolds from the sheaf perspective, a reference: 
![](attachments/image_2021-04-25-01-54-21.png)

## Random Algebraic Topology

> Reference: paper on "constructive" algebraic topology
[J. Rubio, F. Sergeraert / Bull. Sci. math. 126 (2002) 389-412 403](https://www-fourier.ujf-grenoble.fr/~sergerar/Papers/Constructive-AT.pdf)

- Many constructions in algebraic topology can be organized as solutions of fibration problems.

- What are [Quillen equivalence](Quillen%20equivalence.md)? 
	#todo/questions
 	These need to preserve the [model structure](model%20structure) on each side presumably.

- More fundamental: how *should* one prove an [equivalence of categories](equivalence%20of%20categories.md) in general? 
  	#todo/questions
 
- Finding [Unsorted/adjoint (categorical)](Unsorted/adjoint%20(categorical).md) is usually easy, because checking isomorphisms on hom sets is concrete.
  
- If you just have a random functor, does it even *have* right or left adjoints in general? 
	There must be theorems about this. See [adjoint functor theorem](adjoint%20functor%20theorem.md).
  	#todo/questions

- What is the [Stiefel manifold](Stiefel%20manifold.md)? 
 	#todo/questions
	- I should write down an explicit set-theoretic description somewhere. This is definitely in Fomenko.

## 19:38

- Is there a natural exact sequence associated to a [composition series](composition%20series)? 
  	#todo/questions
	- This seems like it should be super easy, we have quotients everywhere.
	- Is there a precise relation to [iterated extensions](iterated%20extensions)..? 
  	#todo/questions