Overview
This research introduces a mathematical concept called two-sided Ehresmann semigroupoids. These newly defined structures are shown to correspond with a particular class of categories, which the researchers term local biordered Ehresmann categories. This established correspondence serves as a unified generalization of the Ehresmann-Schein-Nambooripad Theorem.
Research Context
The work builds upon the existing Ehresmann-Schein-Nambooripad Theorem, a significant result within the theory of semigroups and categories. The goal was to provide a generalization that encompasses both inverse semigroupoids and Ehresmann semigroups. The established correspondence aims to unify previous results within this broader framework.
Approach
The core of the approach involved defining a new mathematical entity: the two-sided Ehresmann semigroupoid. Following this definition, the researchers established a relationship, or correspondence, between these semigroupoids and a specific type of category they named local biordered Ehresmann categories. This correspondence facilitates the generalization of the Ehresmann-Schein-Nambooripad Theorem.
Findings
- Introduction of the notion of two-sided Ehresmann semigroupoids.
- Demonstration of a correspondence between two-sided Ehresmann semigroupoids and local biordered Ehresmann categories.
- This correspondence provides a unified generalization of the Ehresmann-Schein-Nambooripad Theorem for both inverse semigroupoids and Ehresmann semigroups.
- Identification of two-sided restriction semigroupoids as a distinguished subclass within two-sided Ehresmann semigroupoids.
- For the case of two-sided restriction semigroupoids, the associated class of categories was described, extending earlier results concerning restriction semigroups.