One setting for all: metric, topology, uniformity, approach structure
Abstract: For a complete lattice V which, as a category, is monoidal
closed, and for a suitable Set-monad T we consider both (T,V)-algebras
and (T,V)-proalgebras, in generalization of Lawvere's presentation of metric
spaces and Barr's presentation of topological spaces. In this lax-algebraic
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 main 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.
