A Grothendieck topos can be understood as a generalisation of a topological space in which the "points" of the "space" may admit internal structure of their own. For example, given a coherent first-order theory T, one may generalise from the Stone space of complete theories extending T to the topos of all models of T - the so-called classifying topos of the theory. This allows a much finer-grained analysis of the underlying geometry.

However, there are some subtle mismatches between the theory of toposes and the theory of topological spaces; moreover, it is not always clear why a particular construction in topos theory should be the right generalisation of one from general topology. These two points may lead one to question whether toposes really deserve the title of "generalised spaces". The aim of this talk is to define a new notion - that of ionad -which fits the description much more closely.