An Application of Idempotent Monads and Comonads to Compactifications and Unitizations

TL;DR

This paper explores the use of idempotent monads and comonads to establish categorical equivalences between compactifications of spaces and unitizations of C*-algebras, extending classical results to noncommutative settings.

Contribution

It introduces a novel application of monads and comonads to relate compactifications and unitizations, generalizing classical topological and algebraic dualities to noncommutative frameworks.

Findings

Established an equivalence between reflective and coreflective subcategories using monads and comonads.

Extended classical dualities from commutative to noncommutative C*-algebras.

Provided a categorical framework unifying 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.