REU Meeting - 2025-07-08

This following is a brief summary of our research meeting on 2025-07-08.

Meeting summary

We chatted about some of the exercises in Mac Lane, especially Exercise 3c in Section 1.3 and Exercise 6 in Section 1.4. Our main focus was getting a feel for functors between categories, which always consist of two pieces of information: 1) an object function; and 2) an arrow function. We noted how functors need to be compatible with composition and send identity arrows to identity arrows. We also noted how one consequence of those properties is that functors always send invertible arrows to invertible arrows.

We used Example 6 in Section 1.4 as an illustration of categories that are equivalent but not isomorphic. We also briefly talked about natural transformations (and natural isomorphisms) between functors, but we didn't get too nitty gritty. We can return to natural transformations more later when/if we need them.

Tasks for next meeting

• In Categories for the Working Mathematician:
  • Read Section 1.5 and the notes at the end of Chapter 1
  • Try the following exercises:
    • 1.5: 1-6
• In The Joy of Cats:
  • Read/skim through Chapters 5, 7 and 8. There's a lot to digest in those chapters, but we can't meet again late July (or early August), so take your time!

References

Adamek-Herrlich-Strecker - The Joy of Cats: Sections 5, 7 and 8
Mac Lane - Categories for the Working Mathematician: Section 1.5