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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 28th 2013
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   Author: Urs
   Format: MarkdownItexAt _[[differential cohesion]]_ there used to be the statement that every object $X$ canonically has a "spectrum" given by $(Sh_{\mathbf{H}}(X), \mathcal{O}_X)$, but the (simple) argument that $\mathcal{O}_X$ indeed satisfies the axioms of a structure sheaf used to be missing. I have now added it [here](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#StructureSheafIsIndeedStructureSheaf).

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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 28th 2013
   • (edited Dec 28th 2013)
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   Author: zskoda
   Format: MarkdownItexIn 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)

   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)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 28th 2013
   • (edited Dec 28th 2013)
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   Author: Urs
   Format: MarkdownItexI 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 $(\mathcal{X}, \mathcal{O}_X)$ be a [[structured (infinity,1)-topos]] exhibited by a classifying geometric morphism $$ \mathcal{X} \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\longrightarrow}} \mathbf{H} $$ where $\mathbf{H} \coloneqq Sh(\mathcal{G})$ for $\mathcal{G}$ the given [[geometry (for structured (infinity,1)-toposes)]]. Then let $\mathcal{G}^{inf}$ be an infinitesimal thickening, notably the tangent $\infty$-category $\mathcal{G}^{inf} = (T \mathcal{G}^{op})^{op}$ and $\mathbf{H}_{th} \coloneqq Sh(\mathcal{G}^{inf})$. Then an $\mathcal{O}_X$-module $N$ is classified by a lift $$ \array{ \mathcal{X} &\stackrel{N}{\leftarrow}& \mathbf{H}_{th} \\ & {}_{\mathllap{\mathcal{O}_X}}\nwarrow & \downarrow \\ && \mathbf{H} } \,. $$ The observation is that up to this point this is all naturally axiomatized in [[differential cohesion]]. The remaining step is to say that $N$ is quasi-coherent as an $\mathcal{O}_X$-module if it exhibits $\mathcal{X}$ as a $\mathcal{G}^{inf}$-scheme. I am not fully sure yet how to nicely formalize this with differential cohesion.

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

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

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

   Then an $\mathcal{O}_X$-module $N$ is classified by a lift

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

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

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

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeDec 29th 2013
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   Author: zskoda
   Format: MarkdownItexInteresting, I should study this in much more detail.

   Interesting, I should study this in much more detail.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2013
   • (edited Dec 29th 2013)
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   Author: Urs
   Format: MarkdownItexI have added [here](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#FormallyEtaleMorphismInducesEtaleMorphismOfStructuredToposes) the statement and the (simple) proof of the fact that for $f \colon Y \longrightarrow X$ a formally étale morphism in differential cohesion, the induced morphism of étale toposes $(f^\ast \dashv f_\ast) \colon (Sh(Y), \mathcal{O}_Y) \longrightarrow (Sh(X), \mathcal{O}_X)$ is an étale morphism of structured $\infty$-toposes.

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

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeAug 22nd 2014
   • (edited Aug 22nd 2014)
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   Author: Urs
   Format: MarkdownItexI 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.]

   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.]