Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Abstract

Given a regular Gumm category $\mathcal C$ such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory $\mathcal X$ in $\mathcal C$ gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category $\mathcal C$, we first characterize the coverings of the Galois structure $\bar{\Gamma}_1$ induced by the subcategory $\mathcal C_{Ab}$ of the abelian objects in $\mathcal C$. Then we consider $\mathcal C$ as a subcategory of the category $Eq(\mathcal C)$ of the equivalence relations in $\mathcal C$, and we characterize the coverings of the corresponding Galois structure $\bar{\Gamma}_2$. By composing the Galois structures $\bar{\Gamma}_1$ and $\bar{\Gamma}_2$ we obtain the Galois structure $\bar{\Gamma}$ induced by $\mathcal C_{Ab}$ as a subcategory of $Eq(\mathcal C)$. We give the characterization of the $\bar{\Gamma}$-coverings in terms of the coverings of $\bar{\Gamma}_1$ and $\bar{\Gamma}_2$.