I am having a look at this conference by Bertrand Toen about Grothendieck's work.
At 1:14:30 and after, Toen presents the new objects emerging from topos theory in algebraic geometry. He takes the following example: one wants to solve the algebraic equation $$X = 0$$ He says that in a classical framework, it just gives us the solution $0$, so it's a point. Not that interesting. But with the topos point of view, it gives different solutions depending on where we solve the equation: over $\mathbb C$ it gives a point (as before), but over a finite field $\mathbb F_p$ it gives a point together with a (kind of) symmetry. (He represents this last object as a point and loop-edge on the point.)
Could anyone explain to me in an simple manner (as intuitive as possible) the differences Toen is trying to explain?
Thanks in advance