I was reading about topos theory and in many ways some people said that TT can be used to unify logic and geometry. What does that mean?
I have an OK background in category theory (at least up to adjunction theorems) and topology (general, differential and a little bit of algebraic topology) so feel free to those concepts in your explanation.
Best Answer
In topos theory, there is a way to give forcing proofs that use kinds of topologies and sheaves on the topos. So you can create proofs like Cohen’s forcing proofs in topos theory, and it follows the pattern of creating sheaves in geometry.
There is a good book, “Sheaves in Geometry and Logic: A First Introduction to Topos Theory” by Mac Lane and Moerdijk which starts from the basics, so it feels fairly complete.
Topos theory also models intuitionist logic - you can have an intuitionist non-Boolean topos, and, again using a kind of “sheaf,” form a Boolean topos, so that the Boolean topos is sort of a quotient.