Background: I have very little intuition about category theory but I’m trying to understand the motivation behind it.
In mathematics, a topos [...] is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are in a sense a generalization of point-set topology.
Moreover, this video explains that a topos is a “mathematical universe” within which one can do math. I interpret this as saying that different topoi are “foundations of mathematics”. Perhaps the category of sets is an example of such a topos?
I am confused by this: topology is a specific mathematical theory, whereas set theory can be seen as a foundation of mathematics. It seems to me that set theory is far more general than topology, so I don’t understand why the structure of topology would be usable as a “mathematical universe” within which things like groups and linear algebra can be understood.