Morita Equivalence for Many-Sorted Enriched Theories

Morita equivalence detects the situation in which two different theories admit the same class of models for the given theories. We generalise the result of Adámek, Sobral and Sousa concerning Morita equivalence of many-sorted algebraic theories. This generalisation is two-fold. We work in an enriched setting, so the result is parametric in the choice of enrichment. Secondly, the result works for a reasonably general notion of a theory: the class of limits in the theory can be varied. As an example of an application of our result, we show enriched and many-sorted Morita equivalence results, and we recover the known results in the ordinary case.