I am reading here and there about basic synthetic differential geometry. One of the central ideas seems to be that it should be developed in a suitable topos, hence, in particular, a cartesian closed category. I also remember reading Lawvere was especially insistent that cartesian closedness is a crucial bit of structure.

I've also been reading about some basic algebraic geometry and scheme theory. Many of the basic ideas in synthetic differential geometry seem very applicable to algebraic geometry, i.e small objects, tangents, infinitesimal extensions. These feel very geometric. However, neither the category of commutative rings nor the category of affine schemes are cartesian closed, so things like the tangent bundle don't seem to carry over.

A comment to this question does seem to say that restricting to a subcategory of "small" affine schemes does yield cartesian closedness says:

If you restrict to the category of $\Bbbk$-algebras of finite dimension over $\Bbbk$, then the answer is yes, since $\sf Hom$ scheme constructions work well for flat proper maps.

I don't understand the explanation because I don't know enough algebraic geometry, but I would like to know whether we really need to work with schemes over a field, or can we use synthetic differential geometry for $\Bbbk$ a general ring.

I guess what I'm really asking is:

What is the cartesian closed category (topos, really) in which we should do differential geometry of schemes, and where can I read more about this?

My hope is that the synthetic theory will provide the geometric intuition behind the impenetrable wall of formalism I see in most books. If we identify schemes with their functors of points then perhaps we can say our topos is the topos of sheaves on the big Zariski (?) site? Is the category of schemes actually cartesian closed (as a Grothendieck topos)?

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    $\begingroup$ Your guess in the second to last sentence is correct-at least that's one example of a topos which contains scheme theory. Not every sheaf is a scheme, though, and the category of schemes is not cartesian closed! $\endgroup$ – Kevin Carlson Mar 3 '16 at 17:21
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    $\begingroup$ You may want to take a close look here. $\endgroup$ – Andrea Gagna Mar 3 '16 at 22:12

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