Ionads

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.