In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by , who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a scheme. An isocrystal is a crystal up to isogeny. They are -adic analogues of -adic étale sheaves, introduced by and (though the definition of isocrystal only appears in part II of this paper by ). Convergent isocrystals are a variation of isocrystals that work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories. A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals. Crystals over the infinitesimal and crystalline sites The infinitesimal site has as objects the infinitesimal extensions of open sets of . If is a scheme over then the sheaf is defined by = coordinate ring of , where we write as an abbreviation for an object of . Sheaves on this site grow in the sense that they can be extended from open sets to infinitesimal extensions of open sets. A crystal on the site is a sheaf of modules that is rigid in the following sense: for any map between objects , ; of , the natural map from to is an isomorphism. This is similar to the definition of a quasicoherent sheaf of modules in the Zariski topology. An example of a crystal is the sheaf . Crystals on the crystalline site are defined in a similar way. Crystals in fibered categories In general, if is a fibered category over , then a crystal is a cartesian section of the fibered category. In the special case when is the category of infinitesimal extensions of a scheme and the category of quasicoherent modules over objects of , then crystals of this fibered category are the same as crystals of the infinitesimal site. References (letter to Atiyah, Oct. 14 1963) Algebraic geometry