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arXiv · 2503.10868

From Semantics to Syntax: A Type Theory for Comprehension Categories

Niyousha Najmaei·Niels van der Weide·Benedikt Ahrens·Paige Randall North
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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-L\"of 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 $\Pi$-, $\Sigma$-, and identity types. We discuss how it can be viewed as an extension of Martin-L\"of 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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Niyousha Najmaei, Niels van der Weide, Benedikt Ahrens, Paige Randall North. 2025-03-13. From Semantics to Syntax: A Type Theory for Comprehension Categories. https://arxiv.org/abs/2503.10868

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