Duality for noncommutative frames

TL;DR

This paper characterizes noncommutative frames derived from sheaves on topological spaces and establishes dualities linking these structures to sheaves on dissolution locales, extending classical duality theories.

Contribution

It introduces a duality framework for noncommutative frames, connecting them to sheaves on topological and dissolution locales, and generalizes existing duality results.

Findings

Characterization of left-handed noncommutative frames from sheaves

Establishment of dual equivalences of categories

Extension of duality results for skew lattices

Abstract

We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame $A$ arises from a sheaf on the dissolution locale associated to the commutative shadow of $A$. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices in arXiv:1206.5848.