Extending graded comonad semantics to dependent types

Develop a formal extension of the graded-comonad semantics used for simply-typed settings—interpreting well-typed terms with grades as morphisms D^p A → B—to full dependent type theory, making precise how such a semantics accommodates dependent types and judgments.

Background

In simply-typed settings, graded usage can be modeled via a graded comonad D, interpreting terms as morphisms whose domains are decorated by grades. This offers a clear categorical account of resources.

The authors explicitly state that they do not know how to extend this semantics to the general dependent setting. A satisfactory dependent generalization would provide a semantic foundation unifying graded usage and dependency, potentially aligning with quantitative categories with families or other dependent categorical frameworks.