Ionads

TL;DR

This paper introduces ionads, a new mathematical structure that generalizes topological spaces similarly to how toposes relate to locales, expanding the conceptual framework of topology.

Contribution

The paper defines ionads and explores their fundamental properties and relationship with toposes, offering a new perspective in categorical topology.

Findings

Ionads generalize topological spaces.

Ionads relate to toposes similarly as toposes relate to locales.

Foundational aspects of ionads are developed.

Abstract

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.