Category semantics for fusion and refinement of multi-sorted specifications

The goal of this paper is to describe the theory of refinements of specifications based on localizations of categories. The approach allows us to enlarge the family of refinements (i.e. specification morphisms) of the co-complete category Spec - the category of first order theories (specifications) of multi-sorted algebras. We prove that the class of specification morphisms in the category Spec can be enriched by the class of all interpretations of theories from Spec in all definitional extensions of theories of multi-sorted algebras. it may provide a guide for finding a path leading from a given specification to a specification which is a provably correct code in a programming language (like C ++,Lisp, Java)..