2024-10-21

This following is a very brief summary of what happened in class on 2024-10-21.

We finally directly addressed the motto "It's all about the arrows"^[1] 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'll pick back up with this statement (and its proof) next time, when we'll also see lots of examples of this in action.

Concepts

• Universal Properties III - Yoneda's Lemma

References

• Mac Lane: Chapter III, Section 2