Modules over Quantaloids: Applications to the Isomorphism Problem in Algebraic Logic and π-institutions

We solve the isomorphism problem in the context of abstract algebraic logic and of π-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We prove that these categories are strongly complete, strongly cocomplete, and (Epi,Mono)-structured. We prove a duality property, a characterization of monos in virtue of Yoneda Lemma, and as a consequence of this and of the duality, a characterization of epis. We introduce closure operators and closure systems on modules over quantaloids, and its associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators, and as a consequence that Epi = RegEpi, and by duality Mono = RegMono. This is fundamental in the proof of the strong amalgamation property. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. From this we obtain that categories of modules over (small) quantaloids have enough injectives and projectives. Finally, we explain how every π-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of π-institutions.