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It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for (Ì, V)-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that V has a canonical (Ì, V)-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of (Ì, V)-categories. (2000): 18A05, 18D15, 18D20, 18B35, 18C15, 54E15, 54E50. Mathematics Subject ClassificationKey words: V-category, bimodule, monad, (Ì, V)-category, completeness. IntroductionLawvere in his 1973 paper Metric spaces, generalized logic, and closed categories formulates a notion of complete V-category and shows that for (generalised) metric spaces it means Cauchy completeness. This notion of completeness deserved the attention of the categorical community, and the notion of Cauchy-complete category, or Freyd-Karoubi complete category is well-known, mostly in the context of Ab-enriched categories. However, it never got the attention of the topological community. In this paper we interpret Lawvere's completeness in topological settings. We extend Lawvere's notion of complete V-category to the (topological) setting of (Ì, V)-categories (for a symmetric and unital quantale V), and show that it encompasses well-known notions in topological categories, meaning weakly sober space in the category of topological spaces and continuous maps, weakly sober approach space in the category of approach spaces and non-expansive maps, and Cauchy-completeness in the category of quasi-uniform spaces and uniformly continuous maps.We present also a first step towards a possible construction of completion. Indeed, in the setting of V-categories, it is well-known that the completion of a V-category may be built out of the Yoneda embedding X → V X op . In the (Ì, V)-setting, we could prove that V has a canonical (Ì, V)-categorical structure and that every (Ì, V)-category X has a canonical dual * The authors acknowledge partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT and Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT. 1 X op . Using this structure and the free Eilenberg-Moore algebra structure |X| on T X, we get two "Yoneda-like" morphismsFor the latter one we prove a Yoneda Lemma (see 4.2). Furthermore, we show that, under suitable conditions, V is a Lawvere-complete (Ì, V)-category, a first step towards a completion construction which will be the subject of a forthcoming paper.In order to make the presentation of this paper smoother, in Section 1 we recall the notions and properties of V-categories we will generalize throughout. First we introduce V-categories and V-bimodules, and de...

Abstract. For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T, V)-algebras and introduce (T, V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this laxalgebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T, V)-algebras and of (T, V)-proalgebras turn out to be topological over Set. (2000): 18C20, 18B30, 54E15. Mathematics Subject Classifications

In this paper we investigate Cauchy completeness and exponentiablity for quantale enriched categories, paying particular attention to probabilistic metric spaces.

Abstract. For a quantale V, first a closure-theoretic approach to completeness and separation in V-categories is presented. This approach is then generalized to T-categories, where T is a topological theory that entails a set monad Ì and a compatible Ì-algebra structure on V.

For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations of exponentiable objects and morphisms in the category of V-categories and apply them to some well-known examples. In the case V = R + these characterizations of exponentiable morphisms and objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable complete metric spaces are the complete totally convex ones.

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