Enriched categories and many-valued preorders: Categorical, semantical, and topological perspectives

A question from programming arises: if bitstring x compares with bitstring y to some degree α, and if bitstring y satisfies predicate a to some degree β, then how should the possibility be mathematically modeled that bitstring x satisfies predicate a to at least some degree related to both α and β? Mathematically modeling this question is surprisingly intricate when the underlying conjunctions, e.g., of predicates, are non-commutative. Potential applications of this question occur in data-mining, a field in which pattern-matching is important and commonly used. This paper uses ideas from enriched categories over monoidal categories to address such issues and enable pattern-matching techniques to extend to many-valued contexts equipped with non-commutative conjunctions: first, we consider the notion of a set enriched by a po-monoid L, which turns out to be a set equipped with an L-valued preorder; second, we extend the notion of enriched functors to formulate the appropriate definition of “variable-basis morphisms” between preordered sets over different lattice-theoretic bases; third, we construct the notion of topological systems enriched with many-valued preorders and use their extent spaces to motivate and formulate enriched (or preordered) topological spaces—the compatibility or enrichment axioms for such systems and spaces model the programming question stated above using non-commutative tensor products. En route are determined a number of related notions and results: many-valued antisymmetry characterizes the L-T0 separation axiom; many-valued preorders are categorically topological, and many-valued partial orders are monotopological; enriched topological spaces form a topological category; a large inventory of example classes is provided, including programming examples and an extensive discussion of examples based on the L-spectra of complete po-groupoids; and each of these related developments is provided a suitable lattice-theoretic and categorical foundation.