From Semantics to Syntax
A Type Theory for Comprehension Categories
Niyousha Najmaei (École polytechnique)
Niels Van der Weide (Radboud Universiteit Nijmegen)
Benedikt Ahrens (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Paige Randall North (Universiteit Utrecht)
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Abstract
Recent models of intensional type theory have been constructed in algebraic weak factorization systems (AWFSs). AWFSs give rise to comprehension categories that feature non-trivial morphisms between types; these morphisms are not used in the standard interpretation of Martin-Löf type theory in comprehension categories. We develop a type theory that internalizes morphisms between types, reflecting this semantic feature back into syntax. Our type theory comes with Π-, Σ-, and identity types. We discuss how it can be viewed as an extension of Martin-Löf type theory with coercive subtyping, as sketched by Coraglia and Emmenegger. We furthermore define semantic structure that interprets our type theory and prove a soundness result. Finally, we exhibit many examples of the semantic structure, yielding a plethora of interpretations.
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