
Type theories in category theory
We introduce basic notions in category theory to type theorists, includi...
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Syntactic categories for dependent type theory: sketching and adequacy
We argue that locally Cartesian closed categories form a suitable doctri...
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General Semantic Construction of Dependent Refinement Type Systems, Categorically
Refinement types are types equipped with predicates that specify precond...
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Callbyname Gradual Type Theory
We present gradual type theory, a logic and type theory for callbyname...
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Identity types and weak factorization systems in Cauchy complete categories
It has been known that categorical interpretations of dependent type the...
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Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types
We report on a formalization of schemes in the proof assistant Isabelle/...
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A Categorical Programming Language
A theory of data types based on category theory is presented. We organiz...
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Dialectica models of type theory
We present two Dialecticalike constructions for models of intensional MartinLöf type theory based on Gödel's original Dialectica interpretation and the DillerNahm 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 finitelycomplete extensive categories. The second avoids extensivity assumptions using biproducts in a Kleisli category for a fibred additive monad.
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