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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2013
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   Author: Urs
   Format: MarkdownItexThe 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: * _[[formally étale morphism of schemes]]_

   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:

   • formally étale morphism of schemes
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2013
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   Author: Urs
   Format: MarkdownItexFollowing the recent discussion, I have now improved the discussion by replacing the reflection $CRing_{reduced} \hookrightarrow CRing$ with the reflection $CRing_{fin} \hookrightarrow CRing_{fin}^{ext}$ * [Characterization by reduction modality](http://ncatlab.org/nlab/show/formally+étale+morphism+of+schemes#CharacterizationByReductionModality)

   Following the recent discussion, I have now improved the discussion by replacing the reflection $CRing_{reduced} \hookrightarrow CRing$ with the reflection $CRing_{fin} \hookrightarrow CRing_{fin}^{ext}$

   • Characterization by reduction modality
3. • CommentRowNumber3.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 27th 2013
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   Author: Marc Hoyois
   Format: MarkdownItexThat looks good! Is there still a reason to restrict to finitely generated rings now? It doesn't seem to be used anywhere.

   That looks good! Is there still a reason to restrict to finitely generated rings now? It doesn’t seem to be used anywhere.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2013
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   Author: Urs
   Format: MarkdownItexI thought I'd stick to it to keep the site small.

   I thought I’d stick to it to keep the site small.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 28th 2013
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   Author: Marc Hoyois
   Format: MarkdownItexAh 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 $Fun(CRing^{ext}, Set)$.

   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 $Fun(CRing^{ext}, Set)$.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeNov 29th 2013
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   Author: DavidRoberts
   Format: MarkdownItex@Marc - or do you mean small presheaves on the large site?

   @Marc - or do you mean small presheaves on the large site?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 29th 2013
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   Author: Urs
   Format: MarkdownItexHm, 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.

   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.

8. • CommentRowNumber8.
   • CommentAuthorMarc Hoyois
   • CommentTimeNov 29th 2013
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   Author: Marc Hoyois
   Format: MarkdownItex@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”.

   @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”.