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Guitart states in "Toute theorie est algebrique et topologique" as Proposition 17 that the category $\mathbf{Sch}$ of schemes is the category of models of a large mixed sketch. Presumably, it is built using Burroni's sketch of $\mathbf{Top}$. Here is a translation of the relevent paragraph which gives a sketch of the proof.

"This [the functorial approach to schemes] being recalled, starting from the sketch for topological spaces (Prop. 13), we can effectively continue and obtain a mixed sketch whose models are schemes. In other words, like the idea of topology, everything which goes into the definition of schemes, the ideas of rings, local rings, of affine neighborhoods, all of this can therefore be sketched out, in terms of projective and inductive limits."

Questions.

  1. Can someone explain more how that sketch for $\mathbf{Sch}$ looks like?
  2. Why is even the category of affine schemes (i.e. $\mathbf{CRing}^{\mathrm{op}}$) modeled by a large sketch? It seems that Guitart does not make a distinction between $\mathbf{CRing}^{\mathrm{op}}$ and $\mathbf{CRing}$, which is kind of...sketchy. It is only clear to me that $\mathbf{CRing}^{\mathrm{op}}$ is the category of $\mathbf{Set}^{\mathrm{op}}$-valued models of a finite colimit sketch, but we want $\mathbf{Set}$-valued models. Since $\mathbf{Set}^{\mathrm{op}}$ is modelled by a large limit sketch (since it's monadic over $\mathbf{Set}$, see also here), I wonder if the same might be true for $\mathbf{CRing}^{\mathrm{op}}$?
  3. A related question is if the categories $\mathbf{RS}$ and $ \mathbf{LRS}$ of (locally) ringed spaces can be modelled with large sketches, is this possible? I suspect that for this we first need to find a categorical characterization of local homeomorphisms ($\equiv$ sheaves) internal to $\mathbf{Top}$?
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  • $\begingroup$ The idea of Burroni described here is quite general, but not general enough for it to be obvious to me whether manifold-like structures (i.e. structures defined by the existence of an atlas) are within its scope. $\endgroup$
    – Zhen Lin
    May 18 at 6:16
  • $\begingroup$ @ZhenLin Yes, and because of this I also wondered if the statement is true at all. A different issue is that for large sketches there might be a difference between the diagrammatic definition as in Barr-Wells and the functorial definition as in Adamek-Rosicky. Guitart's resp. Burroni's idea only covers the first definition. (But I still don't have a copy of Burroni's thesis, so I cannot tell for sure.) $\endgroup$ May 18 at 6:38
  • $\begingroup$ To the extent that I can decipher Guitart's rather flowery French, I think he's saying that the sketch for schemes is via the functor of points. That is, he's asserting that there's a sketch with underlying category the category of affine schemes. The limit part of the sketch says that Zariski descent is satisfied. The colimit part should say that the functor has a cover by open affines. I don't understand how this latter part is colimit-sketchable, though. $\endgroup$
    – Tim Campion
    May 18 at 15:12
  • $\begingroup$ @TimCampion I think he refers to the geometric definition. I have added some details about that in my question. $\endgroup$ May 18 at 18:16

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