Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 25 1 2026 07 01 When Freudenthal coincides with the smallest compactification with a categorical slant 106968 10.48308/cgasa.2024.234534.1468 EN Simo S Mthethwa Disciplines of Mathematics, School of Agriculture and Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa 0000-0003-1809-5899 Gugulethu Nogwebela Disciplines of Mathematics, School of Agriculture and Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa Journal Article 2024 01 21 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 <em>F</em>-maps, and provide a proof demonstrating that the category of compact regular frames and <em>F</em>-maps forms a coreflective full subcategory of the category of rim-compact frames and <em>F</em>-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. https://cgasa.sbu.ac.ir/article_106968_77c49dcdb7b4cc58e5b07267c2df8908.pdf