2025-10-16 1

This following is a very brief summary of what happened in class on 2025-10-17.

We finally directly addressed the motto "It's all about the arrows." In a given category $C$, we fixed an object $c$ and considered the functor $H_c={\mathrm{Hom}}_C(c,-)$. Working through the details, we saw that this functor was the object function of a functor $H:C^{\text{op}}\to {\mathbf{\text{Set}}}^C$. We then asked the reasonable question:

The first step towards the answer was Yoneda's Lemma, which characterizes the arrows from $H_c$ to other functors. We briefly sketched part of the proof, but ran out of time before hitting the key insight. Alas! We'll save that tidbit for next time.

Concepts

Universal Properties III - Yoneda's Lemma

References

• Mac Lane, Categories for the Working Mathematician: Chapter III, Section 2