What is classified by the (big) crystalline topos?

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?

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The fppf-local rings need to be at least strict Henselian (since the fppf topology is finer than the étale topology, and it makes sense that their residue fields would be algebraically closed (since the fppf-closure of a field is its algebraic closure). Also, I asked a similar question a few weeks ago mathoverflow.net/questions/42258 . –  Harry Gindi Nov 22 '10 at 3:31
@Harry: Thanks for providing the link to your related question. –  Marc Nieper-Wißkirchen Nov 30 '10 at 7:10