REU Meeting - 2025-08-20

This following is a brief summary of our research meeting on 2025-08-20.

Meeting summary

We spent our meeting going through the tasks set at the last meeting, which were meant to slowly build up to a Yoneda-based justification of the definition of monic arrow. As a quick summary, we:

• a natural transformation $\tau :F\Rightarrow G$ between two functors $F,G:C\to \mathbf{S}\mathbf{e}\mathbf{t}$ is called a natural injection if the family of set maps ${\tau }_c:F(c)\to G(c)$ are all injective.
• We described the natural transformation $H_f:H_a\Rightarrow H_b$ that's induced by the "Yoneda functor" $H:C\to {\mathbf{S}\mathbf{e}\mathbf{t}}^C$, whenever we have an arrow $f:a\to b$ in $C$. In short, the natural transformation can be described as the "compose with $f$" maps on arrows.
• We then showed that the condition for $H_f$ to be a natural injection is exactly the condition for $f$ to be monic.

Tasks for next meeting

Coming soon!

References