  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.
