J. Goodrick, B. Kim, and A. Kolesnikov introduced a notion of amenable
collection of functors to a proper category C to develop some simplicial
homology theory in a category. An amenable collection of functors satisfies
good extension and localization properties. A typical example comes from
model theory by considering the category of small subsets of a fixed
monster model with partial elementary embeddings.
Given an amenable collection, we define homology groups and under
(n+1)-complete amalgamation, the n-th homology group is just a set of
homology classes of n-chains of a certain simple form, called n-shell.
Specially, the first homology group is always given by homology classes of
By classifying all possible minimal 2-chains having 1-shell boundaries, we
can compute the first homology groups. In model theory, the first homology
group of a strong type of a model is the abelianization of Lascar group.
For a given abstract group G, we get an amenable collection of functors by
considering G-action on itself by left multiplication. In this case, the
first homology group is the abelianization of G.