Quantalic spectra of semirings

TL;DR

This paper generalizes classical spectra constructions to localic semirings, introducing a spectrum that is valid constructively and can be interpreted as a locale or quantale, enriching the algebraic-topological correspondence.

Contribution

It defines a spectrum for commutative localic semirings, generalizing several classical spectra, and introduces a quantic spectrum with differential information, applicable in topos theory.

Findings

Constructive spectrum valid in elementary toposes.

Spectrum generalizes Stone, Zariski, Gelfand, Hofmann-Lawson spectra.

Introduces quantic spectrum as a commutative quantale.

Abstract

Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a commutative localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains `differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory.