2025-09-18

This following is a very brief summary of what happened in class on 2025-09-18.

We spent the majority of our time discussing the themes of the class, especially category theory. Some highlights include:

• We're in the endgame now (at least for algebra and abstract thinking in general)
• Our main algebraic focus will be on a new algebraic structure, called a module. This structure will generalize the structures of abelian groups and vector spaces.
• We will be interweaving our algebra (module theory) with category theory
• Category theory will help us organize and clarify our thinking, lead us to new insights and unify diverse concepts across much of mathematics

We gave a definition for category and started talking about examples, but we'll have many more examples tomorrow (as well as formally define maps between categories, called functors).

Concepts

• Categories

References

• Mac Lane, Categories for the Working Mathematician: pp. 10–12