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An Application of Idempotent Monads and Comonads to Compactifications and Unitizations

Jeri Ann Spiker

TL;DR

The paper develops an abstract categorical framework using idempotent monads and comonads to realize a maximal-normal equivalence between reflective and coreflective subcategories. It then applies this framework to two intertwined settings: compactifications of locally compact Hausdorff spaces and unitizations of non-unital $C^*$-algebras, using contravariant dualities to transfer results between the contexts. In the topological setting, one-point compactifications yield a reflective subcategory while Stone–Čech compactifications yield a coreflective one, both forming an equivalent pair; in the algebraic setting, Gelfand duality induces corresponding structures on the category of unitizations, producing minimal and maximal unitizations as equivalent subcategories. The noncommutative generalization extends these constructions directly within $\mathbf{U}$, establishing that maximal-normal equivalence persists and that the related adjoint equivalences mirror the commutative case, thereby illustrating a broad categorical mechanism linking compactifications and unitizations.

Abstract

This paper uses monads and comonads to establish a certain type of equivalence between two subcategories, one reflective and one coreflective, in a category whose objects represent compactifications of non-compact locally compact Hausdorff spaces. The equivalence is then examined in the dual category of unitizations of non-unital commutative $C^*$-algebras and subsequently generalized to the noncommutative case.

An Application of Idempotent Monads and Comonads to Compactifications and Unitizations

TL;DR

The paper develops an abstract categorical framework using idempotent monads and comonads to realize a maximal-normal equivalence between reflective and coreflective subcategories. It then applies this framework to two intertwined settings: compactifications of locally compact Hausdorff spaces and unitizations of non-unital -algebras, using contravariant dualities to transfer results between the contexts. In the topological setting, one-point compactifications yield a reflective subcategory while Stone–Čech compactifications yield a coreflective one, both forming an equivalent pair; in the algebraic setting, Gelfand duality induces corresponding structures on the category of unitizations, producing minimal and maximal unitizations as equivalent subcategories. The noncommutative generalization extends these constructions directly within , establishing that maximal-normal equivalence persists and that the related adjoint equivalences mirror the commutative case, thereby illustrating a broad categorical mechanism linking compactifications and unitizations.

Abstract

This paper uses monads and comonads to establish a certain type of equivalence between two subcategories, one reflective and one coreflective, in a category whose objects represent compactifications of non-compact locally compact Hausdorff spaces. The equivalence is then examined in the dual category of unitizations of non-unital commutative -algebras and subsequently generalized to the noncommutative case.
Paper Structure (6 sections, 42 theorems, 51 equations)

This paper contains 6 sections, 42 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compactifications
  4. Unitizations of Commutative $C^*$-Algebras
  5. Unitizations of $C^*$-Algebras
  6. Conclusion

Key Result

Lemma 2.2

Let $(\operatorname{N}, \eta)$, $\mathcal{N}$, and $\operatorname{N}^\eta$ be as given above. Then $\mathrm{Inc}_\mathcal{N}: \mathcal{N} \to \mathcal{C}$ is left adjointable and $\operatorname{N}^\eta$ is a left adjoint of $\mathrm{Inc}_\mathcal{N}$. Thus, $\mathcal{N}$ is a reflective subcategory

Theorems & Definitions (88)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 78 more