On normal subgroupoids

TL;DR

This paper explores algebraic properties of subgroupoids and normal subgroupoids, defining concepts like normalizer, center, and commutator, and extends group theory results to groupoids, including inner isomorphisms and quotient structures.

Contribution

It introduces the notion of normalizer, center, and commutator for subgroupoids, and generalizes the concept of inner automorphisms and quotient groups from groups to groupoids.

Findings

Normalizers are the greatest wide subgroupoids where a subgroupoid is normal.

Centers and commutators are normal subgroupoids with specific properties.

Inner isomorphisms form a normal subgroupoid isomorphic to the quotient by the center.

Abstract

In this paper we present some algebraic properties of subgroupoids and normal subgroupoids. We define the normalizer of a wide subgroupoid $\mathcal{H}$ and show that, as in the case of groups, the normalizer is the greatest wide subgroupoid of the groupoid $\mathcal{G}$ in which $\mathcal{H}$ is normal. Furthermore, we give the definition of center and commutator and prove that both are normal subgroupoids, the first one of the union of all the isotropy groups of $\mathcal{G}$ and the second one of $\mathcal{G}$. Finally, we introduce the concept of inner isomorphism of $\mathcal{G}$ and show that the set of all the inner isomorphisms of $\mathcal{G}$ is a normal subgroupoid, which is isomorphic to the quotient groupoid of $\mathcal{G}$ by its center $\mathcal{Z}(\mathcal{G})$, which extends to groupoids a well-known result in groups.