Crystal (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 07/10/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Crystal_(mathematics)


00:00:29 Crystals over the infinitesimal and crystalline sites
00:00:59 Crystals in fibered categories



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8843454338814605
Voice name: en-US-Wavenet-C


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
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. 
There are several variations of crystals, as follows:

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.