On Diers theory of Spectrum II: Geometries and dualities
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.
Keywords
Cite
@article{arxiv.2012.02167,
title = {On Diers theory of Spectrum II: Geometries and dualities},
author = {Axel Osmond},
journal= {arXiv preprint arXiv:2012.02167},
year = {2021}
}