# 2021-04-26

## Random Notes

Some random notes: #todo

-Working out relative homology, an example:

• Chain of implications for module properties:

• Definitions of common matrix groups:

• Good example of exact triangles:

• Manifolds from the sheaf perspective, a reference:

## Random Algebraic Topology

Reference: paper on “constructive” algebraic topology J. Rubio, F. Sergeraert / Bull. Sci. math. 126 (2002) 389-412 403

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

• What are Quillen equivalence? #todo/questions These need to preserve the model structure on each side presumably.

• More fundamental: how should one prove an equivalence of categories in general? #todo/questions

• Finding Unsorted/adjoint (categorical) 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. #todo/questions

• What is the Stiefel manifold? #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? #todo/questions
• This seems like it should be super easy, we have quotients everywhere.
• Is there a precise relation to iterated extensions..? #todo/questions