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Theory and Applications of Categories, Vol. 32, No. 41, 2017, pp. 1454{1484. TOPOLOGICAL PROPERTIES OF NON-ARCHIMEDEAN APPROACH SPACES EVA COLEBUNDERS AND KAREN VAN OPDENBOSCH Abstract. In this paper we give an isomorphic description of the category of non- Archimedian approach spaces as a category of lax algebras for the ultrafilter monad and an appropriate quantale. Non-Archimedean approach spaces are characterised as those approach spaces having a tower consisting of topologies. We study topological properties p; for p compactness and Hausdorff separation along with low-separation properties, regularity, normality and extremal disconnectedness and link these properties to the condition that all or some of the level topologies in the tower have p. A compactification technique is developed based on Shanin's method. 1. Introduction In monoidal topology [Hofmann, Seal, Tholen (eds.), 2014] starting from the quantale op P+ = [0; 1] ; +; 0 where the natural order ≤ of the extended real halfline [0; 1] is reversed so that 0 = > and 1 = ?, based on Lawvere's description of quasi-metric spaces as categories enriched over the extended real line [Lawvere 1973], one obtains an isomorphism between the category qMet of extended quasi-metric spaces with non- expansive maps and P+ -Cat, the category of P+ -spaces (X; a), where X is a set and a : X−!7 X is a transitive and reflexive P+ -relation, meaning it is a map a : X × X ! P+ satisfying a(x; y) ≤ a(x; z) + a(z; y) and a(x; x) = 0 for all x; y; z in X; with morphisms 0 0 0 f :(X; a) ! (Y; b) of P+ -Cat, maps satisfying b(f(x); f(x )) ≤ a(x; x ) for all x; x in X: Our interest goes to the larger category App of approach spaces and contractions, which contains qMet as coreflectively embedded full subcategory and which is better behaved than qMet with respect to products of underlying topologies. A comprehensive source on approach spaces and their vast field of applications is [Lowen, 2015] in which several equivalent descriptions of approach spaces are formulated in terms of (among others) distances, limit operators, towers and gauges. A first lax-algebraic description of approach spaces was established by Clementino and Hofmann in [Clementino, Hofmann, 2003]. The construction involves the ultrafilter monad = (β; m; e), the quantale P+ and an extension of the ultrafilter monad to P+ -relations. The so constructed category ( ; P+ )-Cat and its isomorphic description App became an important example in the development of monoidal topology [Hofmann, Seal, Tholen (eds.), 2014]. More details on the construction will be Received by the editors 2017-07-17 and, in final form, 2017-11-01. Transmitted by Walter Tholen. Published on 2017-11-13. 2010 Mathematics Subject Classification: 18C15, 18C20, 54A05, 54B30, 54E99. Key words and phrases: Lax algebra, quantale, non-Archimedean approach space, quasi-ultrametric space, initially dense object, topological properties in ( ; P_ )-Cat, compactification. c Eva Colebunders and Karen Van Opdenbosch, 2017. Permission to copy for private use granted. 1454 TOPOLOGICAL PROPERTIES OF NON-ARCHIMEDEAN APPROACH SPACES 1455 recalled in section 2. In this paper we consider the non-Archimedian counterparts of the previous well known constructions. These are obtained by switching the quantale to P_ . As before we consider the extended real halfline [0; 1] as a complete lattice with respect to the natural order ≤ and we reverse its order, so that 0 = > is the top and 1 = ? is the bottom element. Since [0; 1]op is a chain, it is a frame, and we may consider it a quantale with its meet operation which, according to our conventions on the reversed order, is the supremum with respect to the natural order of [0; 1] and as in approach theory this operation will be denoted by _ (note that here we deviate from the notation in [Hofmann, Seal, Tholen (eds.), 2014]). As stated in exercise III.2.B of [Hofmann, Seal, Tholen (eds.), 2014] the u category P_ -Cat is isomorphic to the category qMet of extended quasi-ultrametric spaces (X; d) with non-expansive maps, where an extended quasi-metric is called an extended quasi-ultrametric if it satisfies the strong triangular inequality d(x; z) ≤ d(x; y) _ d(y; z); for all x; y; z 2 X. In section 2, using the ultrafilter monad = (β; m; e), with P_ as quantale and an extension of the ultrafilter monad to P_ -relations we show that the category of associated lax algebras ( ; P_ )-Cat is isomorphic to the reflectively embedded full subcategory NA-App of App; the objects of which are called non-Archimedean approach spaces. In section 3 we give equivalent descriptions of non-Archimedean approach spaces in terms of distances, limit operators, towers and gauges. The characterization in terms of towers can be easily expressed: an approach space is non-Archimedean if and only if its tower (T")"2R+ consists of topologies. We show that NA-App can be generated as the concretely reflective hull by one particular extended quasi-ultrametric. Following the general philosophy to consider lax algebras as spaces, it is our main purpose in this paper to study topological properties p like compactness and Hausdorff separation, along with low-separation properties, regularity, normality and extremal dis- connectedness for non-Archimedean approach spaces. In section 6 we introduce these T properties ( ; P_ )-p as an application to ( ; P_ )-Cat of the corresponding ( ; V)-properties for arbitrary (T; V)-Cat as developed in V.1 and V.2 of [Hofmann, Seal, Tholen (eds.), 2014]. For each of the properties p we characterise ( ; P_ )-p in the context of NA-App. On the other hand we make use of the well known meaning of these properties in the setting of Top and for a non-Archimedean approach space X with tower of topologies + (T")"2R we compare the property ( ; P_ )-p to the properties X has p at level 0, meaning the topological space (X; T0) has p in Top;X almost strongly has p, meaning (X; T") has R+ p in Top for every " 2 0 and X strongly has p, meaning (X; T") has p in Top for every " 2 R+: For p the Hausdorff property, low separation properties or regularity the condi- tions strongly p, almost strongly p and ( ; P_ )-p are all equivalent and different from p at level 0: For p compactness strongly p is equivalent to p at level 0, and almost strongly p is equivalent to ( ; P_ )-p. For p normality or extremal disconnectedness strongly p implies almost strongly p which implies ( ; P_ )-p. Moreover strongly p implies p at level 0. We give counterexamples showing that there are no other valid implications. In the last sections of the paper we develop a compactification technique and prove 1456 EVA COLEBUNDERS AND KAREN VAN OPDENBOSCH that every non-Archimedean approach space X given by its tower of topologies (T")"2R+ can be densely embedded in a compact (at level 0) non-Archimedean approach space Y given by its tower of topologies (S")"2R+ in such a way that at level 0 the compact space (Y; S0) is the Shanin compactification of (X; T0). We give necessary and sufficient conditions for the compactification to be Hausdorff at level 0 and we prove that a space X that is Hausdorff at level 0, ( ; P_ )-regular and ( ; P_ )-normal has such a Hausdorff compactification. 2. P_ -Cat and ( ; P_ )-Cat We follow definitions and notations from [Hofmann, Seal, Tholen (eds.), 2014], unless stated otherwise. In this section we adapt the constructions of P+ -Cat and ( ; P+ )-Cat in [Hofmann, Seal, Tholen (eds.), 2014] to the quantale P_ . Consider the extended real halfline [0; 1] which is a complete lattice with respect to the natural order ≤. We reverse its order, so that 0 = > is the top and 1 = ? is the bottom element. When working with ([0; 1]op; ≤op) and forming infima or suprema we will denote these by infop; Vop; supop or Wop. It means that we will deviate from the conventions made in [Hofmann, Seal, Tholen (eds.), 2014] since we will use both the symbols inf and V when forming infima and sup and W when forming suprema, referring to the natural order on ([0; 1]; ≤). op Since [0; 1] is a chain, it is a frame, and we may consider it a quantale P_ = [0; 1]op; _; 0 with its meet operation (which, according to our conventions, is the supre- mum with respect to the natural order of [0; 1] and will be denoted by _). The map a _ (−): P_ −! P_ is left adjoint to the map a−•(−): P_ −! P_ defined by 0 a ≥ b; a−•b = inffv 2 [0; 1] j b ≤ a _ vg = b a < b: The category P_-Rel has sets as objects and P_ -relations as morphisms. A P_ -relation r : X−!7 Y from X to Y is represented by a map r : X × Y ! P_ . Two P_ -relations r : X−!7 Y and s : Y −!7 Z can be composed in the following way (s · r)(x; z) = inf r(x; y) _ s(y; z) j y 2 Y : 2.1. Definition. P_ -Cat is the category of P_ -spaces (X; a), where X is a set and a : X−!7 X is transitive and reflexive P_ -relation, meaning it is a map a : X × X ! P_ satisfying a(x; y) ≤ a(x; z) _ a(z; y) and a(x; x) = 0 for all x; y; z in X. A morphism f :(X; a) ! (Y; b) of P_ -Cat is a map satisfying b(f(x1); f(x2)) ≤ a(x1; x2) for all x1; x2 in X.
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