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Toposes of Topological Monoid Actions

Morgan Rogers

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.

Toposes of Topological Monoid Actions

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.

Paper Structure

This paper contains 30 sections, 74 theorems, 40 equations.

Table of Contents

  1. Properties of categories of continuous monoid actions
  2. Necessary clopens
  3. Recap on supercompactly generated toposes
  4. Corollaries for toposes of topological monoid actions
  5. The joint covering property
  6. Monoids with topologies
  7. Powersets and inverse image actions
  8. Action topologies
  9. Powder monoids
  10. Prodiscrete monoids, nearly discrete groups
  11. The canonical surjective point
  12. Equivariant relations and congruences
  13. Endomorphisms of the canonical point
  14. Bases of congruences
  15. Factorizing topologies
  16. ...and 15 more sections

Key Result

Lemma 1.2

Let $M$ be a monoid equipped with a topology $\tau$ and $X$ an $M$-set. Then $X$ is an $(M,\tau)$-set if and only if for each $x \in X$ and $p \in M$, the set is open in $\tau$. We call the collection of all such $\mathcal{I}_x^p$ the necessary clopens for $X$.

Theorems & Definitions (197)

  • Remark 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • proof
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • proof
  • Remark 1.6
  • ...and 187 more