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

  • 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.


  • 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
Links to this page
#web/quick-notes #todo #todo/questions