Dialectica models of type theory
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We present two Dialectica-like constructions for models of intensional Martin-Löf type theory based on Gödel's original Dialectica interpretation and the Diller-Nahm variant, bringing dependent types to categorical proof theory. We set both constructions within a logical predicates style theory for display map categories where we show that 'quasifibred' versions of dependent products and universes suffice to construct their standard counterparts. To support the logic required for dependent products in the first construction, we propose a new semantic notion of finite sum for dependent types, generalizing finitely-complete extensive categories. The second avoids extensivity assumptions using biproducts in a Kleisli category for a fibred additive monad.
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