Functorial Polar Functions

Abstract

W∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W∞ and its dual a functorial covering function on CmpT2,∞. We demonstrate that functorial polar functions give rise to reflective hull classes in W∞ and that functorial covering functions give rise to coreflective covering classes in CmpT2,∞. We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal α, the class of α-projectable ℓ-groups is reflective in W∞, and the class of α-disconnected compact Hausdorff spaces is coreflective in CmpT2,∞. Lastly, the notion of a functorial polar function (resp. functorial covering function) is generalized to sublattices of polars (resp. sublattices of regular closed sets)