We present two Dialectica-like constructions for models of intensional Martin-L\"of type theory based on G\"odel'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.

中文翻译：

---

类型论的方言模型

我们基于G'odel的原始Dialectica解释和Diller-Nahm变体，提出了两种类型的Intental Martin-L \“类型理论模型的类似于Dialectica的构造，从而将依赖类型引入了分类证明理论。我们在显示地图类别的逻辑谓词样式理论中设置了两种构造，在这些理论中，我们证明了依赖产品和Universe的“准混合”版本足以构建其标准对应物。为了在第一个结构中支持从属乘积所需的逻辑，我们为从属类型提出了一个新的有限和语义概念，即对有限完成的广泛类别进行了概括。第二种方法避免了将纤维状添加剂一元单体使用Kleisli类别中的双产品的可扩展性假设。