# 2024-10-01

This following is a very brief summary of what happened in class on 2024-10-02.

We began by looking at a few more examples of functors, specifically the following two functors from

- the functor
, which on objects maps each commutative ring to the group of units (i.e., multiplicatively invertible elements). We noted that the arrow map simply takes each ring morphism to the group morphism that is identical on elements. (We mentioned that it's a fact every ring morphism takes invertible elements to invertible elements, so the map does indeed make sense.) - the functor
, which on objects maps each commutative ring to the group of invertible matrices.

These two functors will be connected (next week) via the determinant map, which is a natural transformation.

We then switched gears and introduced the idea of a "universal property." We will return to this idea several times over the next few weeks, gradually filling in the details. For now, we spent our time looking at a list of examples, some familiar and some less so.

## Concepts

## References

- Mac Lane: Pages 13-15 and selected content from Chapter III