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Classical multi-sorted equational theories and their free algebras have beenfundamental in mathematics and computer science. In this paper, we present ageneralization of multi-sorted equational theories from the classical($Set$-enriched) context to the context of enrichment in a symmetric monoidalcategory $V$ that is topological over $Set$. Prominent examples of suchcategories include: various categories of topological and measurable spaces;the categories of models of relational Horn theories without equality,including the categories of preordered sets and (extended) pseudo-metricspaces; and the categories of quasispaces (a.k.a. concrete sheaves) on concretesites, which have recently attracted interest in the study of programminglanguage semantics.  Given such a category $V$, we define a notion of $V$-enriched multi-sortedequational theory. We show that every $V$-enriched multi-sorted equationaltheory $T$ has an underlying classical multi-sorted equational theory $|T|$,and that free $T$-algebras may be obtained as suitable liftings of free$|T|$-algebras. We establish explicit and concrete descriptions of free$T$-algebras, which have a convenient inductive character when $V$ is cartesianclosed. We provide several examples of $V$-enriched multi-sorted equationaltheories, and we also discuss the close connection between these theories andthe presentations of $V$-enriched algebraic theories and monads studied inrecent papers by the author and Lucyshyn-Wright.

Free algebras of topologically enriched multi-sorted equational theories