REU Meeting - 2025-08-06

This following is a brief summary of our research meeting on 2025-08-06.

Meeting summary

We spent most of the time going over the solutions to exercises 1-6 in Section 1.5 of Maclane. Most of the solutions were straightforward, following almost directly from the definitions (of monic, epi, etc.).

The one big exception was Exercise 5, which was to prove that epis in Grp are group morphisms that are surjective (as set maps). That argument was a proof by contradiction that was a lot more complicated than one might expect, with an elaborate hint that involved considering two different cases. We didn't fill in the details, since we don't care too much about this particular exercise, but the statement is nice to have.

We briefly ended by tiptoeing are way towards Yoneda's Lemma. We are going to take a deep dive into this result next, since it's the original inspiration for the motto "It's all about the arrows!"

Tasks for next meeting

• Reading through the following notes, which slowly build to Yoneda's Lemma:
  • Universal Properties I - Inspiring Examples
  • Universal Properties II - Commutative diagrams, cones and limits
  • Universal Properties III - Yoneda's Lemma

References

Mac Lane - Categories for the Working Mathematician: Chapter III