I will talk about objects connected to generalized amalgamation properties in first-order stable theories. The objects are: a certain group of automorphisms of a model of the theory, vertex groups of polygroupoids definable in a model of the theory, homology group associated with a strong type of the theory. It turns out that, under some natural conditions, the three groups are isomorphic. I will introduce the above objects, explain their connection to generalized amalgamation, and will outline the arguments showing that the groups are isomorphic.