group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Could not include topos theory - contents
Galois cohomology is the group cohomology of Galois groups $G$. Specifically, for $G$ the Galois group of a field extension $L/K$, Galois cohomology refers to the group cohomology of $G$ with coefficients in a $G$-module naturally associated to $L$.
Galois cohomology is studied notably in the context of algebraic number theory.
Galois cohomology of a field $k$ is essentially the étale cohomology of the spectrum $Spec(k)$.
See also at comparison theorem (étale cohomology).
We make some comments on the formulation of Galois cohomology in cohesive homotopy type theory.
As discussed at Galois theory, the absolute Galois group $G_{Galois}$ of a field $K$ is the fundamental group of the spectrum $X \coloneqq Spec(K)$. Hence its delooping $\mathbf{B}G_{Galois}$ is the fundamental groupoid
In cohesive homotopy type theory there exists the fundamental ∞-groupoid-construction – the shape modality $\Pi$ –
Moreover, by the discussion at group cohomology in the section group cohomology - In terms of homotopy type theory
a $G_{Galois}$-module $A$ is a $\mathbf{B}G_{Galois}$-dependent type;
the group cohomology is the dependent product over the function type
Hence, generally in cohesive homotopy type theory, for $X$ a type and
a $\Pi(X)$-dependent type, we can say that the corresponding $\infty$-Galois cohomology is
Warning. Currently there is not written down yet a model for cohesive homotopy type theory given by a cohesive (∞,1)-topos over a site like the étale site.
Jean-Pierre Serre, Galois cohomology, Springer Monographs in Mathematics (1997)
Cohomologie galoisienne, Lecture Notes in Mathematics, 5 (Fifth ed. 1994), Springer-Verlag, MR1324577
Grégory Berhuy, An introduction to Galois cohomology and its applications (pdf)
M. Kneser, Lectures on Galois cohomology of classical groups (pdf)
Wikipedia, Galois cohomology
A generalization in the setup of corings: