We present two Dialectica-like constructions for models of intensionalMartin-L\”of type theory . We propose a new semantic notion of finite sum for dependent types, generalizingfinitely-complete extensive categories . The second avoids extensivityassumptions using biproducts in a Kleisli category for a fibred additive monad . We set both constructions within a logical predicates style theory for display map categories where we show that ‘quasifibred’ versions of dependentproducts and universes suffice to construct their standard counterparts .

Author(s) : Sean K. Moss, Tamara von Glehn

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Keywords : theory - dialectica - constructions - models - type -

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