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.
- Can someone explain more how that sketch for $\mathbf{Sch}$ looks like?
- 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 $\mathbf{CRing}^{\mathrm{op}}$?
- 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 $\mathbf{Top}$?
