I'm not a mathematician, but I have some knowledge in Category theory (mainly from computer science background). As I understand it, Topoi, and in particular presheaf category can "capture" the quintessence of the notion of topological space, and I've heard that this is the key to, in a way, connect continuous and discrete mathematics.
How can I understand this idea more profoundly without getting too abstract? Can you explain in "simple" terms what this "connection" could be? And what it is useful for? (maybe that could be the mathematical framework to unify quantum theory (discrete) and general relativity (continuous))?
For every topological space $X$ we can construct the category of sheaves of sets, $Sh(X)$. This is a topos. A topos requires no extra structure, it's just a category which satisfies some technical requirements.
If $A$ and $B$ are topoi then a morphism of topoi $u: A \rightarrow B$ is a pair of adjoint functors $u^*⊣ u_*$ with $u_*: A \rightarrow B$.
Every continuous function $f:X \rightarrow Y$ induces an adjoint pair $f^*⊣ f_*$ with $f_*: Sh(X) \rightarrow Sh(Y)$, these functors are called the inverse/direct image functors respectively.
If both $X$ and $Y$ are sober topological spaces (soberness is a weak separation axiom) then morphisms of topoi $Sh(X) \rightarrow Sh(Y)$ are in bijection with maps $X \rightarrow Y$.
This shows that the topos $Sh(X)$ captures topological properties of $X$ very nicely.