Crystal (Math)

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In mathematics, crystals are 
Cartesian sections of certain fibered categories.
They were introduced by Alexander Grothendieck (1966a),
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.

WIKI

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https://www.scientificamerican.com/a.../crystal-math/


Crystal Math
By Charles Q. Choi on March 1, 2008

Diamonds are rarities not just on earth but also mathematically.
The crystal structure of diamond has two key distinguishing properties,
notes mathematician Toshikazu Sunada of Meiji University in Japan.

It has maximal symmetry, which means that its components cannot be rearranged
to make it any more symmetrical than it is,
and a strong isotropic property, which means that it looks the same
when viewed from the direction of any edge.

In the February Notices of the American Mathematical Society,
Sunada finds that out of an infinite universe of crystals that
can exist mathematically, just one other shares these properties with diamond.
Whereas diamond is a web of hexagonal rings, its cousin is made of 10-sided rings.

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An isocrystal is a crystal up to isogeny.

They are p-adic analogues of Ql-adic étale sheaves, introduced by Grothendieck (1966a)
and Berthelot and Ogus (1983)
(though the definition of isocrystal only appears in part II of this paper by Ogus (1984)).

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.