In this paper on page 8 there is the following passage:
Let $A$ be an algebra, $M$ an $A$-module. There are the following approaches to the "cohomology of $A$ with coefficients in $M$".
- Abelian cohomology defined as $H^*(Lin(R_*,M))$, where $R_*$ is a resolution of $A$ in the category of $A$-modules.
- Non-abelian cohomology defined as $H^*(Der(\mathfrak{F}_*,M))$, where $\mathfrak{F}_*$ is a resolution of $A$ in the category of algebras and $Der(-,M)$ denotes the space of derivations with coefficients in $M$.
If one takes $R_*$ to be projective resolution, this so-called abelian homology is just the right derived functor of the hom functor, i.e. the Ext-functor modulo an index-shift. Correct?
What does non-abelian cohomology refer to? The paper references Quillen's Homotopical Algebra, but I can't find a explanation of this so-called "non-abelian cohomology". nlab was not too helpful either.