Lascar groups and the first homology groups of strong types in rosy theories

TL;DR

This paper establishes a connection between Lascar groups and first homology groups of strong types in rosy theories, revealing their structure and cardinalities, with explicit examples illustrating non-trivial cases.

Contribution

It introduces a canonical surjective homomorphism from Lascar groups to homology groups in rosy theories and characterizes their kernels, advancing understanding of their algebraic structure.

Findings

First homology groups in rosy theories are either trivial or have size at least continuum.

Explicit examples of rosy theories with non-trivial homology groups are provided.

In these examples, homology groups are isomorphic to their Lascar groups.

Abstract

For a rosy theory, we give a canonical surjective homomorphism from a Lascar group over $A=\acl^{eq}(A)$ to a first homology group of a strong type over $A$, and we describe its kernel by an invariant equivalence relation. As a consequence, we show that the first homology groups of strong types in rosy theories have the cardinalities of one or at least $2^{\aleph_0}$. We give two examples of rosy theories having non trivial first homology groups of strong types over $\acl^{eq}(\emptyset)$. In these examples, these two homology groups are exactly isomorphic to their Lascar group over $\acl^{eq}(\emptyset)$.