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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.
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)
I don’t know if this is in Grothendieck, but I doubt it. The traditional text closest to DAG5 in spirit seems to be
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.
Interesting, I should study this in much more detail.
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.
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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