group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Crystalline cohomology is the abelian sheaf cohomology with respect to the crystalline site of a scheme. Hence, put more generally, it is the cohomology of de Rham spaces/coreduced objects.
Crystalline cohomology serves to refine the notion of de Rham cohomology for schemes.
Crystalline cohomology is in particular a Weil cohomology and is generalized by the notion of rigid cohomology.
Let $X$ be a scheme overy a base $S$. The crystalline site $Cris(X/S)$ of $X$ is
the category whose objects are all nilpotent $S$-immersions $U \hookrightarrow T$, where $U$ is an open set of $X$ and and the ideal on $T$ defining this immersion being endowed with a nilpotent divided power structure (…details…).;
the Grothendieck topology on this category is the Zariski topology.
If $S$ is of characteristic 0, then $Cris(X/S)$ coincides with the infinitesimal site of $X$. (…details…).
Crystalline cohomology of $X$ is the cohomology of the de Rham space of $X$. See there for more.
In differential homotopy type theory the infinitesimal flat modality sends coefficients to coefficients for crystalline cohomology.
Related entries: crystal, infinitesimal site, rigid cohomology, Dieudonné module, Monsky-Washnitzer cohomology, Grothendieck connection
An original account of the definition of the crystalline topos is section 7, page 299 of
A more recent account is
Discussion of this in the modern context of higher geometry/D-geometry is in
A p-adic cohomology for varieties in characteristic $p$ it was it was discussed in
The comparison theorem (crystalline cohomology) shown there (theorem V2.3.2) shows that the crystalline cohomology over $\mathbb{Z}/p$ is canonically identified with the de Rham cohomology of a lift to p-adic geometry if one exists. See also
Pierre Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9
Pierre Berthelot, A. Ogus, $F$-Isocrystals and de Rham Cohomology, I, Invent. math. 72, 1983, pp. 159-199
Discussion of this in terms of Cech cohomology is in
See also