Metric, Topology and Multicategory – A Common Approach

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. Mathematics Subject Classification: 18C20, 18D15, 18A05, 18B30, 18B35.