REU Meeting - 2025-08-11

This following is a brief summary of our research meeting on 2025-08-11.

Meeting summary

We spent the entire meeting talking about Yoneda's Lemma, recalling how to each object $c$ in a category $C$ we can associate a functor $H_{c} = {Hom}_{C} (-, c)$ , which encodes the information about all arrows to $c$ in the category. This association is the first step in creating a full-on functor $H : C \to {S e t}^{C}$ , which on objects sends each object $c \in C$ to the functor $H_{c} : C \to S e t$ described above. Yoneda's Lemma tells us that this functor is a "fully faithful embedding" of the category $C$ into the (much larger) category ${S e t}^{C}$ . This is the original motivation of the motto "It's all about the arrows!"

We then talked a little a bit about the inspiring examples which might lead one to naturally discover Yoneda's Lemma. To get more hands-on practice, though, we will now see how Yoneda's Lemma naturally leads on to the definition of monomorphism.

Tasks for next meeting

Suppose $F, G : C \to S e t$ are two functors from a given category $C$ to the category of sets, and suppose $τ : F \Rightarrow G$ is a natural transformation. Propose a "natural" (pun intended) definition for $τ$ to be called a natural injection.
Suppose $a, b \in C$ are two objects in a given category $C$ , and $f : a \to b$ is an arrow in $C$ . The functor $H$ should provide a natural transformation $H_{f} : H_{a} \Rightarrow H_{b}$ . Describe this natural transformation, i.e., for every object $c \in C$ , describe the set map $H_{a} (c) \to H_{b} (c)$ . (Hint: Write down what those sets are.)
Using your proposed definition for natural injection, show that $f : a \to b$ is a monomorphism in $C$ if and only if the corresponding natural transformation $H_{f} : H_{a} \Rightarrow H_{b}$ is a natural injection.

References

Universal Properties I - Inspiring Examples
Universal Properties III - Yoneda's Lemma
Special morphisms