Stone duality for spectral sheaves and the patch monad

TL;DR

This paper extends classical Stone duality to spectral sheaves by introducing a patch monad, linking spectral space sheaves with distributive skew lattices through a new duality framework.

Contribution

It establishes a duality between spectral sheaves and right distributive bands, extending Stone duality with a novel patch monad and algebra correspondence.

Findings

Duality between spectral sheaves and right distributive bands

Patch monad on sheaves over spectral spaces

Distributive skew lattices as algebras of the patch monad

Abstract

We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices. The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.