The First Homology Group of a G-Set

摘要

We introduce a notion of a geometric C-set, which is a G-set equipped with a closure operation and an independence relation. For a geometric G-set. we define homology groups of orbits and we identify the first homology groups of orbits. These homology groups generalize the homology groups of types in model theory. Also we construct a groupoid on the set of orbit whose vertex groups are the first homology groups of orbits and from this, we associated a groupoid on the Stone space of types whose vertex groups are the first homology groups of types.