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In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are really only about working over $\mathrm{Spec}\mathbb{Z}$.

  1. I have seen (for example on nLab) that $\mathrm{Sh}(\mathrm{CRing}_{fp}^{op})$ is the classifying topos of the theory of local rings. This is nice because then the points of the topos are exactly local rings in $\mathrm{Set}$. I was wondering what exactly does the restricting to finitely presented rings achieve? Is it necessary for proving it is the classifying topos or is it just to bypass size issues? If so...

  2. Let $\mathscr{U}$ be a Grothendieck universe and define $\mathrm{Sh}(\mathrm{CRing}_{\mathscr{U}}^{op})$ as sheaves on the dual category of commutative rings in $\mathscr{U}$ with respect to the Zariski topology. Then is $\mathrm{Sh}(\mathrm{CRing}_{\mathscr{U}}^{op})$ the classifying topos of local rings in $\mathscr{U}$? If not, is it related in some way?

For reference, I am very much an amateur in topos theory so my questions may be ill-formulated. I apologise in advance.

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    $\begingroup$ Not sure if this is helpful, but VIII.5 in Sheaves in Geometry and Logic talks about this. Basically you want your category to be freely generated by $\mathbb{Z}[x]$ which yields f.p. rings. $\endgroup$ Mar 15, 2021 at 21:35
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    $\begingroup$ To get a classifying topos for local rings you need to restrict the site to finitely presented rings. Conveniently, this also solves a size issue. If you allow, say, countably presented rings then you still get an essentially small site but the topos is no longer a classifying topos for local rings. $\endgroup$
    – Zhen Lin
    Mar 15, 2021 at 22:19
  • $\begingroup$ I guess I must looking into the details. Thanks! $\endgroup$
    – user577413
    Mar 17, 2021 at 23:15

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