2009/12/08 by Richard Garner, Garner, Richard · 2 citations
Mathematics · #18B25 #54A05 #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.0912.1415
openalex publication_date 2009/12/08 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The notion of Grothendieck topos may be considered as a generalisation of that of topological space, one in which the points of the space may have non-trivial automorphisms. However, the analogy is not precise, since in a topological space, it is the points which have conceptual priority over the open sets, whereas in a topos it is the other way around. Hence a topos is more correctly regarded as a generalised locale, than as a generalised space. In this article we introduce the notion of ionad, which stands in the same relationship to a topological space as a (Grothendieck) topos does to a locale. We develop basic aspects of their theory and discuss their relationship with toposes.