We will discuss the recently suggested definition of prismatic cohomology that unifies several integral cohomology theories in p-adic geometry. We briefly start with an introduction of prisms as generalization of perfectoid rings, and then describe available constructions of a prismatic site:
1) a variant of the complex $\Delta_{R/A}$$ (2016);
2) construction via topological Hochschild homology (2018).
It requires as input a highly non-trivial calculation from homotopy theory
due to Bockstedt.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/5312