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 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 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.
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Clementino, M.M., Hofmann, D. & Tholen, W. One Setting for All: Metric, Topology, Uniformity, Approach Structure. Applied Categorical Structures 12, 127–154 (2004). https://doi.org/10.1023/B:APCS.0000018144.87456.10
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DOI: https://doi.org/10.1023/B:APCS.0000018144.87456.10