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This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question.

There has been quite a lot of work done in synthetic differential geometry, and recently some work in synthetic algebraic geometry as well. Working synthetically in a topos highlights similarities between different "flavors" of geometry. The question is, can we do geometry synthetically inside a generic "geometric" topos, and get corresponding results in algebraic, differential, etc., all flavors of geometry for free?

I understand that this approach would be somehow limited (after all, DG is different from AG), but I'm interested in how far could one get via this approach. Or is this approach fruitful at all, and if not, why not?

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    $\begingroup$ Yes, we can! For instance, the same internal proof shows that, in AG, the Grassmannian is a smooth scheme over the base, and in DG, the Grassmannian is a smooth manifold (see Section 20.3 of these notes of mine). However, as you say, this approach is limited, the internal world of the toposes used in AG does differ quite a bit from their DG counterpart. For instance, in the toposes for AG, any map $R \to R$ is a polynomial. A generalization of this observation is the basis for synthetic AG, yet totally false for SDG. $\endgroup$
    – Ingo Blechschmidt
    Commented Mar 25, 2021 at 21:20
  • $\begingroup$ @IngoBlechschmidt Thanks Ingo. Your thesis is truly a source of inspiration. I was wondering if we can go as far as things like e.g. differential forms and deformation theory, but I guess that's way, way ahead. $\endgroup$
    – xuq01
    Commented Mar 26, 2021 at 0:51

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