§ Topology is really about computation --- part 2
Here, we're going to describe whatever I've picked up of sheaves in the past
couple of weeks. I'm trying to understand the relationship between sheaves,
topoi, geometry, and logic. I currently see how topoi allows us to model logic,
and how sheaves allow us to model geometry, but I see nothing about the
relationship! I'm hoping that writing this down will allow me to gain some
perspective on this.
§ What is a sheaf?
Let's consider two sets , . Now, given a function
, we can restrict this function to .
So, we get to invert the direction:
We should now try to discover some sort of structure to this "reversal"
business. Perhaps we will discover a contravariant functor! (Spoiler: we will).