TL;DR
This paper classifies Grothendieck topologies on Azumaya algebras using supernatural numbers, explores their sheaf categories, and relates to UHF-algebras and group actions, advancing the understanding of Azumaya toposes.
Contribution
It provides a classification of Azumaya toposes via supernatural numbers and links their sheaf categories to UHF-algebras and group actions, offering new insights into their structure.
Findings
Classification of topologies using supernatural numbers
Description of sheaf toposes related to UHF-algebras
Connection between topos points and projective general linear group actions
Abstract
In arXiv:1606.07885, many different Grothendieck topologies were introduced on the category of Azumaya algebras. Here we give a classification in terms of sets of supernatural numbers. Then we discuss the associated categories of sheaves and their topos-theoretic points, which are related to UHF-algebras. The sheaf toposes that correspond to a single supernatural number have an alternative description, involving actions of the associated projective general linear group.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology