§ monic and epic arrows
This is trivial, I'm surprised it took me this long to internalize this fact.
When we convert a poset into a category, we stipulate that
If we now consider the category of sets and functions between sets,
and arrow is a function from to . If is
monic, then we know that . That is, a monic arrow
behaves a lot like a poset arrow!
Similarly, an epic arrow behaves a lot like the arrow in the inverse poset.
I wonder if quite a lot of category theoretic diagrams are clarified by thinking
of monic and epic directly in terms of controlling sizes.