A duality theorem

Abstract

This chapter presents a duality theorem providing, for each geometric theory, a natural bijection between its geometric theory extensions (also called ‘quotients’) and the subtoposes of its classifying topos. Two different proofs of this theorem are provided, one relying on the theory of classifying toposes and the other, of purely syntactic nature, based on a proof-theoretic interpretation of the notion of Grothendieck topology. Via this interpretation the theorem can be reformulated as a proof-theoretic equivalence between the classical system of geometric logic over a given geometric theory and a suitable proof system whose rules correspond to the axioms defining the notion of Grothendieck topology. The role of this duality as a means for shedding light on axiomatization problems for geometric theories is thoroughly discussed, and a deduction theorem for geometric logic is derived from it.