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?