Real metrics are defined as enriched categories over the extended reals, with linear cases derived from potential functions, as part of weighted algebraic topology.
Mac Lane, Categories for the working mathematician, Springer, 1971
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An expository account that interprets Lorentz manifolds via enriched categories over the extended real line, extending Lawvere's metric spaces.
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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.
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Enriched categories, real metrics and Lorentz manifolds
An expository account that interprets Lorentz manifolds via enriched categories over the extended real line, extending Lawvere's metric spaces.