Free constructions for comprehension categories
2026-07-29 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors study a special kind of mathematical structure called comprehension categories, which help model types and their relationships in logic and computer science. They focus on a smaller group within these, named Lawvere-Ehrhard comprehension categories, and show how to identify them by comparing related structures called fibrations. The authors also explain how to build the most basic comprehension category from a fibration and how to create the simplest Lawvere-Ehrhard comprehension category from a Jacobs comprehension category.
comprehension categoryJacobs comprehension categoryLawvere-Ehrhard comprehension categoryfibrationtype dependencytype morphismfree constructioncategorical modelmorphismtype theory
Authors
Francesco Dagnino, Jacopo Emmenegger, Andrea Giusto
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.