Free constructions for comprehension categories
arXiv:2607.27170
The paper investigates comprehension categories used to model type dependency, introduces a subclass called Lawvere‑Ehrhard comprehension categories, and provides constructions of free comprehension categories over fibrations as well as free Lawvere‑Ehrhard comprehension categories over Jacobs comprehension categories.
Abstract
Jacobs comprehension categories subsume a large class of categorical models of type dependency, supporting also the description of morphisms between types. We study the relationship between comprehension categories and a particular subclass, which we call Lawvere-Ehrhard comprehension categories. First, we characterize this subclass by comparing a fibration of terms and a fibration of type morphisms associated to a given comprehension category. Next, we provide the construction of the free comprehension category over a fibration. Finally, we construct the free Lawvere-Ehrhard comprehension category over a Jacobs comprehension category.