The local universes model: an overlooked coherence construction for dependent type theories

TL;DR

This paper introduces a new coherence theorem for comprehension categories, enabling strict models of dependent type theory with standard constructors, based on weak models with minimal stability assumptions.

Contribution

It provides a method to convert weak comprehension categories into strictly stable models, broadening the applicability of coherence results in dependent type theory.

Findings

Constructs strict models from weak comprehension categories.

Works under minimal stability assumptions, weaker than Beck--Chevalley.

Applicable to homotopy-theoretic models of type theory.

Abstract

We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types. Precisely, we take as input a "weak model": a comprehension category, equipped with structure corresponding to the desired logical constructions. We assume throughout that the base category is close to locally Cartesian closed: specifically, that products and certain exponentials exist. Beyond this, we require only that the logical structure should be *weakly stable* --- a pure existence statement, not involving any specific choice of structure, weaker than standard categorical Beck--Chevalley conditions, and holding in the now standard homotopy-theoretic models of type theory. Given such a comprehension category, we construct an equivalent split one, whose logical structure is strictly stable under reindexing. This yields an interpretation of type theory with the chosen constructors. The model is adapted from Voevodsky's use of universes for coherence, and at the level of fibrations is a classical construction of Giraud. It may be viewed in terms of local universes or delayed substitutions.