Authors: Morgan Rogers ¹

• 1 University of Insubria

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.

Comment: 58 pages. Final version appearing in Compositionality. Project under the INdAM Doctoral Programme in Mathematics and/or Applications Cofunded by Marie Sklodowska-Curie Actions, INdAM-DP-COFUND-2015, grant number 713485

• 18B25 - Topoi
• 18B40 - Groupoids, semigroupoids, semigroups, groups (viewed as categories)
• 18D25 - Actions of a monoidal category, tensorial strength

• 1 zbMATH Open