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 $\mathbf{CRing}$ to $\mathbf{\text{Grp}}$:

• the functor ${\bullet }^{\times }$, which on objects maps each commutative ring $R$ to the group $R^{\times }$ of units (i.e., multiplicatively invertible elements). We noted that the arrow map simply takes each ring morphism $f:R\to S$ to the group morphism $f^{\times }:R^{\times }\to S^{\times }$ that is identical on elements. (We mentioned that it's a fact every ring morphism takes invertible elements to invertible elements, so the map $f^{\times }$ does indeed make sense.)
• the functor ${\mathrm{GL}}_n$, which on objects maps each commutative ring $R$ to the group ${\mathrm{GL}}_n(R)$ of invertible $n\times n$ 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

• Functors
• Universal Properties I - Inspiring Examples

References

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