2024-10-22

This following is a very brief summary of what happened in class on 2024-10-22.

We took a deeper look at Yoneda's Lemma, including sketching out most of the details of the proof. The main takeaway is that each category $C$ can be embedded into the functor category ${\mathbf{\text{Set}}}^C$, by sending each object $c\in C$ to the functor $H_c={\mathrm{Hom}}_C(c,-)$.

This allows us to place any given category $C$ into a (much) larger category of functors. It also allowed us to finally formally define "universal properties." In particular, whenever we have a natural isomorphism $\tau :H_c\stackrel{\sim }{\Rightarrow }F$ for some object $c\in C$ and functor $F\in {\mathbf{\text{Set}}}^C$, we say:

• $\tau$ is a universal property of the object $c$
• the object $c$ (together with the natural isomorphism $\tau$) is a representation of the functor $F$ (and we call $F$ a representable functor)

We then proceeded to give a handful of examples, but there are many, many more!

Concepts

• Universal Properties III - Yoneda's Lemma

References

• Mac Lane: Chapter III, Section 2