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The characterization of formally étale morphisms of schemes by the infinitesimal shape modality had been scattered a bit through the nLab (at Q-category, at formally etale morphism a bit, at differential cohesion a little).
To make the statement more recognizable, I put it into this new entry here:
Following the recent discussion, I have now improved the discussion by replacing the reflection $CRing_{reduced} \hookrightarrow CRing$ with the reflection $CRing_{fin} \hookrightarrow CRing_{fin}^{ext}$
That looks good! Is there still a reason to restrict to finitely generated rings now? It doesn’t seem to be used anywhere.
I thought I’d stick to it to keep the site small.
Ah yes, I didn’t think about that. But this finiteness hypothesis is too strong for practical purposes since it makes any formally étale morphism actually étale. A better solution might be to define the adjoint triple on “large” presheaves $\widehat{PSh}$, and/or to directly define $\int_{inf}$ as an endofunctor of $Fun(CRing^{ext}, Set)$.
@Marc - or do you mean small presheaves on the large site?
Hm, wait, every formally étale morphism in the small site would be étale, but the point is to characterize formally étale morphisms between the presheaves on that site. These need not ve étale.
@DavidRoberts
I assume you mean presheaves that are small colimits of representables. I don’t think it’s clear from the definition that $\int$ or $\flat$ should preserve small presheaves (for instance, is the de Rham space of $Spec(A)$ a small presheaf?). It’s useful that $\int$ is defined on arbitrary functors $CRing^{ext}\to Set$ anyway, since we may not know whether a given moduli problem is “small”.
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