Amalgamation and Interpolation in the Category of Heyting Algebras
نویسنده
چکیده
This is the first of two papers describing how properties of open continuous maps between locales (which are the lattice-theoretic generalisation of topological spaces) can be used to give very straight-forward, constructive proofs of certain properties of first-order intuitionistic theories. The properties we have in mind are those of stability of a conservative interpretation of theories under pushout, and appropriate categorical formulations of Craig’s Interpolation Theorem and the Beth Definability Theorem. It is thus the methods of proof rather than the results themselves that are novel, and we present them in the spirit of a demonstration of the usefulness of a category-theoretic approach to constructive logic. In this paper we will consider only propositional intuitionistic theories and their lattice-theoretic counterpart, Hfzyting algebras. At this level the Interpolation Theorem becomes a statement about free Heyting algebras:
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