Metric, Topology and Multicategory - A Common Approach
Abstract: 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.
