REU Meeting - 2025-07-31

This following is a brief summary of our research meeting on 2025-07-31.

Meeting summary

We chatted about the tasks assigned at the end of our last meeting, and then veered off on a tangent about monoidal categories, monoids in monoidal categories and monads. So we've decided to take a mini-detour to take a quick peek at some of those ideas, partly out of curiosity and partly to have those concepts in the back of our minds as we move forward with our categorization of representation theory.

Tasks for next meeting

• Skim through the first three sections of Chapter VII.
• Familiarize yourself with the basic concept of a monoidal structure and a monoid in a monoidal category.
• Write down some key/familiar examples of categories with a monoidal structure, and examples of monoids in those categories.
• In the special case in which your category is of the form $C^C$, where $C$ is some fixed category (and hence $C^C$ is the category of all functors $F:C\to C$), parse the meaning of a monoid in that monoidal category. Thus define a monad and rejoice!

References

Mac Lane, Categories for the Working Mathematician: Chapter VII