Real metrics are defined as enriched categories over the extended reals, with linear cases derived from potential functions, as part of weighted algebraic topology.
Lawvere, State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions, IMA Preprint Series 84, University of Minnesota, 1984
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Weighted algebraic topology, II (Real valued metrics)
Real metrics are defined as enriched categories over the extended reals, with linear cases derived from potential functions, as part of weighted algebraic topology.