Free algebras of topologically enriched multi-sorted equational theories

Classical multi-sorted equational theories and their free algebras have been fundamental in mathematics and computer science. In this paper, we present a generalization of multi-sorted equational theories from the classical ($Set$-enriched) context to the context of enrichment in a symmetric monoidal category $V$ that is topological over $Set$. Prominent examples of such categories 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-metric spaces; and the categories of quasispaces (a.k.a. concrete sheaves) on concrete sites, which have recently attracted interest in the study of programming language semantics. Given such a category $V$, we define a notion of $V$-enriched multi-sorted equational theory. We show that every $V$-enriched multi-sorted equational theory $T$ has an underlying classical multi-sorted equational theory $\mathrm{\mid }T\mathrm{\mid }$, and that free $T$-algebras may be obtained as suitable liftings of free $\mathrm{\mid }T\mathrm{\mid }$-algebras. We establish explicit and concrete descriptions of free $T$-algebras, which have a convenient inductive character when $V$ is cartesian closed. We provide several examples of $V$-enriched multi-sorted equational theories, and we also discuss the close connection between these theories and the presentations of $V$-enriched algebraic theories and monads studied in recent papers by the author and Lucyshyn-Wright.