Enriched Logical Connections

In the setting of enriched category theory, we describe dual adjunctions of the form $L\dashv R:{\mathsf{Spa}}^{op} \longrightarrow{\mathsf{Alg}}$ between the dual of the category Spa of “spaces” and the category Alg of “algebras” that arise from a schizophrenic object Ω, which is both an “algebra” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of Ω is that of a module that allows to lift optimally the structure of a “space” and an “algebra” to certain diagrams. Our approach allows to give a unified framework known from logical connections over the category of sets and analyzed, e.g., by Hans Porst and Walter Tholen, with future applications of logical connections in coalgebraic logic and elsewhere, where typically, both the category of “spaces” and the category of “algebras” consist of “structured presheaves”.