When Freudenthal coincides with the smallest compactification with a categorical slant

Articles in Press

Document Type : Research Paper

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Disciplines of Mathematics, School of Agriculture and Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

10.48308/cgasa.2024.234534.1468

Abstract

In this note, for a certain class of regular continuous frames, we establish conditions that are equivalent to saying that the Freudenthal compactification and the smallest compactification are indistinguishable; in turn, this expands the list of conditions under which the smallest compactification is perfect, which is available in the literature. We define a new class of morphisms between frames, called F-maps, and provide a proof demonstrating that the category of compact regular frames and F-maps forms a coreflective full subcategory of the category of rim-compact frames and F-maps. This coreflection is evidenced by the join map associated with the Freudenthal compactification. Accordingly, this provides an affirmative answer to the question by Herrlich, which inquired whether the Freudenthal compactification can be regarded as a reflection with "sensible" maps.

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References

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Categories and General Algebraic Structures with Applications

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Available Online from 06 May 2026

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