Toposes of Topological Monoid Actions

TL;DR

This paper shows that categories of continuous actions of topological monoids on discrete spaces form Grothendieck toposes, characterizes these toposes, and explores their Morita-equivalence classes and functorial properties.

Contribution

It provides a comprehensive characterization of toposes arising from topological monoid actions and introduces classes of monoids with good topological properties for Morita-equivalence.

Findings

Categories of monoid actions are Grothendieck toposes.

Characterization of toposes via canonical points.

Identification of powder and complete monoids for Morita-equivalence.

Abstract

We demonstrate that categories of continuous actions of topological monoids on discrete spaces are Grothendieck toposes. We exhibit properties of these toposes, giving a solution to the corresponding Morita-equivalence problem. We characterize these toposes in terms of their canonical points. We identify natural classes of representatives with good topological properties, `powder monoids' and then `complete monoids', for the Morita-equivalence classes of topological monoids. Finally, we show that the construction of these toposes can be made (2-)functorial by considering geometric morphisms induced by continuous semigroup homomorphisms.