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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2013

    At differential cohesion there used to be the statement that every object XX canonically has a “spectrum” given by (Sh H(X),𝒪 X)(Sh_{\mathbf{H}}(X), \mathcal{O}_X), but the (simple) argument that 𝒪 X\mathcal{O}_X indeed satisfies the axioms of a structure sheaf used to be missing. I have now added it here.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 28th 2013
    • (edited Dec 28th 2013)

    In the case of ringed 1-topoi, are the axioms already say in the Grothendieck’s SGA ? (I do not know, I just want to see how far this agrees with 1-categorical picture)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2013
    • (edited Dec 28th 2013)

    I don’t know if this is in Grothendieck, but I doubt it. The traditional text closest to DAG5 in spirit seems to be

    • Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).

    But I haven’t actually looked at it yet.

    On the other hand, the notion of quasicoherent sheaves in DAG 8 Quasi-Coherent Sheaves and Tannaka Duality Theorems certainly reduces to the standard 1-categorical one. And that’s the one I am talking about here.

    If you look at section 2.2 in DAG8, you see that the simple idea expressed there is the following nice constuction:

    let (𝒳,𝒪 X)(\mathcal{X}, \mathcal{O}_X) be a structured (infinity,1)-topos exhibited by a classifying geometric morphism

    𝒳𝒪 XH \mathcal{X} \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\longrightarrow}} \mathbf{H}

    where HSh(𝒢)\mathbf{H} \coloneqq Sh(\mathcal{G}) for 𝒢\mathcal{G} the given geometry (for structured (infinity,1)-toposes). Then let 𝒢 inf\mathcal{G}^{inf} be an infinitesimal thickening, notably the tangent \infty-category 𝒢 inf=(T𝒢 op) op\mathcal{G}^{inf} = (T \mathcal{G}^{op})^{op} and H thSh(𝒢 inf)\mathbf{H}_{th} \coloneqq Sh(\mathcal{G}^{inf}).

    Then an 𝒪 X\mathcal{O}_X-module NN is classified by a lift

    𝒳 N H th 𝒪 X H. \array{ \mathcal{X} &\stackrel{N}{\leftarrow}& \mathbf{H}_{th} \\ & {}_{\mathllap{\mathcal{O}_X}}\nwarrow & \downarrow \\ && \mathbf{H} } \,.

    The observation is that up to this point this is all naturally axiomatized in differential cohesion.

    The remaining step is to say that NN is quasi-coherent as an 𝒪 X\mathcal{O}_X-module if it exhibits 𝒳\mathcal{X} as a 𝒢 inf\mathcal{G}^{inf}-scheme. I am not fully sure yet how to nicely formalize this with differential cohesion.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeDec 29th 2013

    Interesting, I should study this in much more detail.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2013
    • (edited Dec 29th 2013)

    I have added here the statement and the (simple) proof of the fact that for f:YXf \colon Y \longrightarrow X a formally étale morphism in differential cohesion, the induced morphism of étale toposes (f *f *):(Sh(Y),𝒪 Y)(Sh(X),𝒪 X)(f^\ast \dashv f_\ast) \colon (Sh(Y), \mathcal{O}_Y) \longrightarrow (Sh(X), \mathcal{O}_X) is an étale morphism of structured \infty-toposes.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2014
    • (edited Aug 22nd 2014)

    I have edited the text and section outline at differential cohesion and idelic structure a bit more, for readability and flow of the argument (or at least I hope that’s what I did).

    [edit: oh, sorry, this is posted in the wrong thread here. Anyway.]

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