Let p be a prime, W the ring of Witt vectors of a perfect field k of
characteristic p and 味 a primitive pth root of unity. We introduce a
new notion of calculus over W that we call absolute calculus. It may be seen
as a singular version of the q-calculus used in previous work, in the sense
that the role of the coordinate is now played by q itself. We show that what
we call a weakly nilpotent absolute connection on a finite free module is
equivalent to a prismatic vector bundle on W[味]. As a corollary of a
theorem of Bhatt and Scholze, we finally obtain that an absolute connection
with a frobenius structure on a finite free module is equivalent to a lattice
in a crystalline representation. We also consider the case of de Rham prismatic
crystals as well as Hodge-Tate prismatic crystals