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.

💡 Research Summary

The paper “Ionads” introduces a new categorical structure that sits between topological spaces and Grothendieck toposes, filling a conceptual gap left by the fact that a topos generalises locales rather than spaces. In a classical topological space the points are primitive and the open sets are derived as subsets of the point set; in a Grothendieck topos the open‑set side (the locale) is primitive and points are merely separating objects. This asymmetry means that the functor from spaces to toposes is neither full nor faithful, and one cannot recover a space from its topos unless the space satisfies sobriety.

To remedy this, the author defines an ionad as a pair ((X, I_X)) where (X) is a set of points and (I_X : \mathbf{Set}_X \to \mathbf{Set}_X) is a cartesian (finite‑limit preserving) comonad. The coalgebras for (I_X) form a category (\mathcal{O}(X)) which the author calls the “opens” of the ionad. Because (\mathbf{Set}_X) is a topos and coalgebras for a cartesian comonad on a topos are again a topos, (\mathcal{O}(X)) is always a (elementary, cocomplete) Grothendieck topos. Moreover, the canonical geometric morphism (\mathbf{Set}_X \to \mathcal{O}(X)) is surjective, i.e. the points of (X) separate the opens. Consequently an ionad is precisely a topos equipped with a separating set of points; it can be thought of as a “discrete‑point” presentation of a topos, just as a sober space is a “discrete‑locale” presentation of a locale.

The paper then develops a basis theory for ionads, mirroring the classical basis description of a topology. A basis for an ionad on a set (X) is a small category (\mathcal{B}) together with a flat functor (M:\mathcal{B}\to\mathbf{Set}_X). Flatness means that for each point (x\in X) the comma category ((M(-)(x)\downarrow)) is cofiltered. Using the Yoneda embedding (\mathcal{B}\to\mathbf{Set}^{\mathcal{B}^{op}}) one extends (M) to a colimit‑preserving functor (M\otimes-:\mathbf{Set}^{\mathcal{B}^{op}}\to\mathbf{Set}_X). This functor has a right adjoint (