Problem
Research questions and friction points this paper is trying to address.
comprehension categories
Lawvere-Ehrhard comprehension categories
fibrations
free constructions
type dependency
Free constructions for comprehension categories
🤖 AI Summary
This study investigates the systematic construction of free comprehension categories, with a particular focus on Lawvere–Ehrhard comprehension categories, and clarifies their relationship to Jacobs comprehension categories. By comparing term fibrations with type-morphism fibrations, the paper provides the first precise characterization conditions for Lawvere–Ehrhard comprehension categories. It further constructs two kinds of free objects: one yielding a free comprehension category from an arbitrary fibration, and another producing a free Lawvere–Ehrhard comprehension category from a given Jacobs comprehension category. This work extends the categorical semantic foundations of dependent type theories and refines the algebraic framework underlying their models.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
comprehension categories
Lawvere-Ehrhard comprehension categories
free construction
fibrations
type dependency
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