  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 $|T|$,
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 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.
