What is classified by the (big) crystalline topos?

Question

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is the theory it describes? I.e., what type of objects in a topos E correspond to geometric morphisms from E to the big crystalline topos (over some scheme)?

P.S.: One paragraph later, G.C. Wraith adds that he conjectures that the fppf-topos classifies algebraically closed local rings. Has this been established somewhere?

Answer 1

I answered this question in my PhD thesis, supervised by the OP: https://arxiv.org/abs/2206.11244

In the simplest case, the big crystalline topos over $\operatorname{Spec} \mathbb{Z}$ classifies the geometric theory of a local ring $A$ together with a nil ideal $I \subseteq A$ and a PD structure (divided powers structure) on $I$.

In the general affine case (Theorem 5.7.4), we have a pair of base schemes $\operatorname{Spec} R / \operatorname{Spec} K$, where $K$ is a ring equipped with a PD ideal $I_K \subseteq K$ and $R$ is a $K/I_K$ algebra. Then the local ring $A$ in the theory is a $K$ algebra, more precisely, it is equipped with a homomorphism of PD rings $K \to A$, and the quotient ring $A/I$ is equipped with a compatible $R$ algebra structure.

The thesis also explains how to obtain a classified theory in the case of non-affine base schemes (Theorem 3.7.6 for the Zariski topos, Theorem 5.8.3 for the crystalline topos), by "gluing" together geometric theories according to how the base scheme is glued together from affine schemes.

In all these cases, appropriate finiteness conditions have to be imposed on the objects of the site defining the topos for the classification result to hold. For the big Zariski topos, the site contains finitely presented algebras over the base ring, or schemes locally of finite presentation over the base scheme. For the crystalline topos, we need a notion of finite PD presentation.