Welcome to another episode of categories by diagrams! Today we will investigate the connection between adjunctions and monads. The results I present here are quite elementary. However, it is a nice exercise for learning how to utilize string diagrams. Of course, to the trained category theorist this approach will be less efficient than directly applying Yoneda-type arguments. Still, I think the string diagrams let us easily understand the underlying structure (compared to the usual commutative diagrams.)

Today I want to present an alternative way for reasoning about (strict) 2-categories that allows us to express seemingly difficult concepts in a very simple way visually. To see how powerful this approach is, we will study (weak) Kan extensions as a concept in the 2-category of (adequately size bounded) categories.

Here is a small collection of properties that amenable groups enjoy.

Here I want to present a quick and easy proof of the first Sylow theorem. It uses basic combinatorics only and yields a super elementary proof of Cauchy’s theorem for free. The goal is to prove the following statement:

It is time for some recreational linear algebra. Todays’ challenge will be to show that $\GL_n$ is an affine group scheme in the most economical way possible (even if it takes away some algebraic insight.)