Toposes of Topological Monoid Actions
Morgan Rogers

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.
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