On the representation of non-Archimedean objects

Banaschewski [B. Banaschewski, Über nulldimensionale Räume, Math. Nachr. 13 (1955) 129-140], Monna [A.F. Monna, Remarques sur les métriques non-Archimédiennes. I, II, Indag. Math. 53 (1950) 122-133, 179-191], van Rooij [A.C.M. van Rooij, Non-Archimedean uniformities, Kyungpook Math. J. 10 (1970) 21-30; A.C.M. van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, 1978] and others have studied non-Archimedean spaces in several different categories. In the present paper we propose a general categorical approach in order to introduce non-Archimedean objects in arbitrary well-fibred topological categories. We show that these subcategories of non-Archimedean objects can always be represented as coreflective subcategories of the category NA of non-Archimedean spaces (introduced in [D. Deses, E. Lowen-Colebunders, On completeness in a non-Archimedean setting, via firm reflections, Bull. Belg. Math. Soc. (Special volume) (2002) 49-61]). Hence the category of non-Archimedean spaces can be viewed as a supercategory containing the non-Archimedean objects of any well-fibred topological construct. Moreover, in the case where this representation yields a hereditary subcategory of NA, we will be able to show the existence of a firm U-reflective subcategory (in the sense of [G.C.L. Brümmer, E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolin. 33 (1992) 131-147; G.C.L. Brümmer, E. Giuli, H. Herrlich, Epireflections which are completions, Cahiers Topologie Géom. Différentielle Catégoriques 33 (1992) 71-73]) of complete non-Archimedean objects.