On Diers theory of Spectrum II: Geometries and dualities

TL;DR

This paper advances the theory of spectra in categorical contexts by constructing spectra via multi-adjunctions, exploring geometric and duality aspects, and formalizing spectral duality through fibrations and 2-functoriality.

Contribution

It introduces a new construction of spectra associated with multi-adjunctions, incorporating geometric insights and axiomatizing spectral duality via fibrations and 2-category frameworks.

Findings

Constructed spectra as spaces of local units with orthogonality topology

Extended Diers' original construction to multi-adjunction contexts

Axiomatized spectral duality through morphisms of fibrations

Abstract

This second part comes to the construction of the spectrum associated to a situation of multi-adjunction. Exploiting a geometric understanding of its multi-versal property, the spectrum of an object is obtained as the spaces of local units equipped with a topology provided by orthogonality aspects. After recalling Diers original construction, this paper introduces new material. First we explain how the situation of multi-adjunction can be corrected in a situation of adjunction between categories of modeled spaces as in the topos-theoretic approach. Then we come to the 2-functorial aspects of the process relatively to a 2-category of Diers contexts. We propose an axiomatization of the notion of spectral duality through morphisms between fibrations over a category of spatial objects, and show how such situations get back right multi-adjoint functors.