Abstract
The simple concepts of (general) distance function and homometry (a map that preserves distances up to a calibration) are introduced, and it is shown how some natural distance functions on various mathematical objects lead to concrete embeddings of the following categories into the resulting category DIST°: quasi-pseudo-metric, topological, and (quasi-)uniform spaces with various kinds of maps; groups and lattice-ordered abelian groups; rings and modules, particularly fields; sets with reflexive relations and relation-preserving maps (particularly directed loop-less graphs and quasi-ordered sets); measured spaces with Radon-continuous maps; Boolean, Brouwerian, and orthomodular lattices; categories with combined objects, for example topological groups, ordered topological spaces, ordered fields, Banach spaces with linear contractions or linear continuous maps and so on.
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Heitzig, J. Many Familiar Categories can be Interpreted as Categories of Generalized Metric Spaces. Applied Categorical Structures 10, 505–520 (2002). https://doi.org/10.1023/A:1020526912025
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DOI: https://doi.org/10.1023/A:1020526912025