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.

@article{paperbot3871,
  title = {From Semantics to Syntax: A Type Theory for Comprehension Categories},
  author = {Niyousha Najmaei and Niels van der Weide and Benedikt Ahrens and Paige Randall North},
  journal = {Proceedings of the ACM on Programming Languages},
  volume = {10},
  number = {POPL},
  year = {2026},
  doi = {10.1145/3776725}
}