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For a symmetric monoidal-closed category V and a suitable monad T on the category of sets, we introduce the notion of reflexive and transitive (T, V)-algebra and show that various old and new structures are instances of such algebras. Lawvere's presentation of a metric space as a V-category is included in our setting, via the Betti-Carboni-Street-Walters interpretation of a V-category as a monad in the bicategory of V-matrices, and so are Barr's presentation of topological spaces as lax algebras, Lowen's approach spaces, and Lambek's multicategories, which enjoy renewed interest in the study of n-categories. As a further example, we introduce a new structure called ultracategory which simultaneously generalizes the notions of topological space and of category.

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The paper discusses interactions between order and topology on a given set which do not presuppose any separation conditions for either of the two structures, but which lead to the existing notions established by Nachbin in more special situations. We pursue this discussion at the much more general level of lax algebras, so that our categories do not concern just ordered topological spaces, but also sets with two interacting orders, approach spaces with an additional metric, etc. IntroductionContrary to widespread perception, in his beautiful monograph Topology and Order [N2] Nachbin did not formally introduce a notion of topological ordered space, or of ordered topological space. He did introduce normally (pre)ordered and compact ordered spaces, but even the original article [N1] contains no formal definition in the general case, despite the fact that its first paragraph is entitled "On topological ordered spaces". Rather, he simply refers to a topological space equipped with a preorder, which normally is assumed to be closed (as a subset of the product space). About the reasons I can only speculate. But since he often cites the case of the discrete order as the one giving the corresponding ordinary topological notion or result, whereas a topological space with a closed discrete (or any) order must necessarily be Hausdorff, I conclude 1 The author's research work is assisted by an NSERC Discovery Grant

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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

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