To give further background to the question here is an extract from google scholar with a search on "Frohlich homological"' (A. Frohlich was a well known worker in algebraic number theory):
Non‐Abelian Homological Algebra I. Derived Functors and Satellites,
A Fröhlich - Proceedings of the London Mathematical Society, 1961 - Wiley Online Library
The purpose of this forthcoming series of papers is the development of a homology (or
cohomology) theory based on structures with non-commutative addition. The module over a
ring, which was hitherto the basic concept for homological algebra, will here be replaced by …
A search on "Lue homological" also gives relevant articles. One of the points that comes out from his 1971, 1981 articles, derived from Frohlich's work, is that his form of the construction of the cohomology is by classes of representative algebraic objects: for example, in the case of groups these objects are commonly called crossed $n$-fold extensions.
The more usual approach is in terms of classes of cocycles. To my mind, a difficulty here is: what do you do with a cocycle when you have got it? By contrast, all sorts of things can be done with or asked about algebraic objects.
This is relevant to the algebraic objects used in the book Nonabelian Algebraic Topology, where, following J.H.C. Whitehead, his crossed modules, and what are now called crossed complexes, and other structures, are used to model some homotopy types. One aspect of the approach
is that the complications of, say, a 3-cocycle definition, are, following work of J. Huebschmann, put into the differentials of a "standard free crossed resolution", so that the 3-cocycle becomes a morphism of crossed complexes; there are many homotopical methods for constructing such morphisms.
To give further background, note that in a letter dated 02/05/1983 Alexander Grothendieck wrote:
Don’t be surprised by my supposed efﬁciency in digging out the right
kind of notions – I have just been following, rather let myself be pulled
ahead, by that very strong thread (roughly: understand non commutative
cohomology of topoi!) which I kept trying to sell for about ten or twenty
years now, without anyone ready to ``buy” it, namely to do the work. So
ﬁnally I got mad and decided to work out at least an outline by myself.
These were his ideas in "Pursuing Stacks".
So I think there is still a lot to take in, assess, and evaluate!