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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2013

    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:

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2013

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

    • CommentRowNumber3.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 27th 2013

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

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2013

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

    • CommentRowNumber5.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 28th 2013

    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 PSh^\widehat{PSh}, and/or to directly define inf\int_{inf} as an endofunctor of Fun(CRing ext,Set)Fun(CRing^{ext}, Set).

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 29th 2013

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

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2013

    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.

    • CommentRowNumber8.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 29th 2013

    @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)Spec(A) a small presheaf?). It’s useful that \int is defined on arbitrary functors CRing extSetCRing^{ext}\to Set anyway, since we may not know whether a given moduli problem is “small”.