Pointfree pointwise convergence, Baire functions, and epimorphisms in truncated archimedean ℓ-groups

Abstract

We define pointfree pointwise convergence, and use it to define the Baire functions on a locale. The main result is that the Baire functions on a locale coincide with the continuous functions on its P-locale coreflection. Furthermore, we show that the Baire functions on a locale constitute the epicompletion of the continuous functions in the relevant category.The relevant category is T, the category of truncated archimedean ℓ-groups, hereafter nicknamed truncs. T is closely related to the famous category W of unital archimedean ℓ-groups. The universal objects in T are of the form R0L, the trunc of real-valued locale maps L→R which vanish at the designated point of a pointed locale L.We provide an intuitive definition of pointwise convergence in R0L which extends the classical definition, and show that it has a number of nice properties: all homomorphisms and operations of T are pointwise continuous, and a pointwise dense extension is a trunc epimorphism. Conversely, we show that every epic extension G→H has an epic extension H→K such that G is pointwise dense in K.We show that the rich theory of epimorphisms in W carries over to T with only minor modification. In particular, the epicomplete truncs comprise a full monoreflective subcategory, and are characterized as those objects of the form R0P for a P-locale P. In light of these facts, a reformulation of the last clause of the preceding paragraph is that any trunc is pointwise dense in any epicompletion. And a trunc is epicomplete iff it is pointwise complete, i.e., has no proper extension in which it is pointwise dense.Finally, for a given pointed locale L, we define the functions of Baire class α on L in the classical fashion. A function is Baire class 0 if it lies in R0L, and of Baire class β if it is the pointwise limit of a sequence of functions of Baire class α<β. A Baire function on L is a function of Baire class α for some α. Our results can be summarized as follows.TheoremFor a pointed locale L with P-locale coreflectionP⁎L→L, the Baire functions on L are precisely the continuous functions onP⁎L, i.e., those ofR0P⁎L.TheoremThe embeddingR0L→R0P⁎Lis the functorial epicompletion inT.