2 research outputs found
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
(-enriched) context to the context of enrichment in a symmetric monoidal
category that is topological over . 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 , we define a notion of -enriched multi-sorted
equational theory. We show that every -enriched multi-sorted equational
theory has an underlying classical multi-sorted equational theory ,
and that free -algebras may be obtained as suitable liftings of free
-algebras. We establish explicit and concrete descriptions of free
-algebras, which have a convenient inductive character when is cartesian
closed. We provide several examples of -enriched multi-sorted equational
theories, and we also discuss the close connection between these theories and
the presentations of -enriched algebraic theories and monads studied in
recent papers by the author and Lucyshyn-Wright.Comment: 51 pages plus six page Appendix. Revised to include more discussion
of enrichment of categories of algebras (expanded Remarks 3.1.4 and 4.1.6;
added more details to Section 6; expanded item 6.9; added an extra item to
Theorem 6.10; shortened Remark 6.11