Interpolation in Grothendieck Institutions

Grothendieck institutions have recently emerged as an important mathematical structure underlying heterogenuous multi-logic specification. On the other hand, interpolation properties of logics underlying specification formalisms play an important role in the study of structured specifications.

In this paper we solve the interpolation problem for Grothendieck institutions. Our main result can be used in the applications in several different ways. It can be used to establish interpolation properties for multi-logic Grothendieck institutions, but also to lift interpolation properties from unsorted logics to their many sorted variants. The importance of the latter resides in the fact that, unlike other structural properties of logics, many sorted interpolation is a non-trivial generalization of unsorted interpolation.

The concepts, results, and the applications discussed in this paper are illustrated with several examples from conventional logic and algebraic specification theory.

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