What Is a Derived Signature Morphism?

The notion of signature morphism is basic to the theory of institutions. It provides a powerful primitive for the study of specifications, their modularity and their relations in an abstract setting. The notion of derived signature morphism generalises signature morphisms to more complex constructions, where symbols may be mapped not only to symbols, but to arbitrary terms. The purpose of this work is to study derived signature morphisms in an institution-independent way. We will recall and generalize two known approaches to derived signature morphisms, introduce a third one, and discuss their pros and cons. We especially study the existence of colimits of derived signature morphisms. The motivation is to give an independent semantics to the notion of derived signature morphism, query and substitution in the context of the Distributed Ontology, Modeling and Specification Language DOL.