I was reading the Russian translation of Harvests and Seeds and in the footnote where Grothendieck lists his 12 contributions (including diets), we find the following lines:
Из этих тем наиболее обширной по своей значимости мне представляется тема топосов, которая осуществляет идею синтеза алгебраической геометрии, топологии и арифметики.
Now, I'm not Russian, but I translate this as follows "In my opinion, among these subjects, the largest, in terms of importance, is that of topoi, which implements the idea of synthesis of geometry algebraic, topology and arithmetic. "
I am aware that topoi allows us to efficiently pack some pieces of homological algebra (which I would call). This can be used to establish some properties of the normal cohomology, for example, useful in arithmetic geometry. Another type of applications that we find in mathematical logic. I do not know them very well, but I've heard that forcing the use of topoi is an interesting prospect.
The question is: do we have independent evidence that Grothendieck's estimate of the importance of the topoi is accurate?
To clarify, consider the diagrams. They are rather necessary if you want to have clear conceptual proofs, for example, of Weil's conjectures or of Mazur's theorem on torsion points. You can try to prove the first in particular cases in a purely computer-based way (from the definition equations) and you will realize that this problem is difficult. I think it's not unreasonable to expect that you can try something similar with this last one. So, there are schema applications that can be expressed in very simple language and you can convince yourself that trying to prove them in very simple language is not going to fly. That's why, as a hard worker, I have tremendous respect for projects.
Do we have something like this for topoi? Are there topoi applications that could be "masked" in the same way?