Strictification of weakly stable type-theoretic structures using generic contexts

by   Rafaël Bocquet, et al.

We present a new strictification method for type-theoretic structures that are only weakly stable under substitution. Given weakly stable structures over some model of type theory, we construct equivalent strictly stable structures by evaluating the weakly stable structures at generic contexts. These generic contexts are specified using the categorical notion of familial representability. This generalizes the local universes method of Lumsdaine and Warren. We show that generic contexts can also be constructed in any category with families which is freely generated by collections of types and terms, without any definitional equality. This relies on the fact that they support first-order unification. These free models can only be equipped with weak type-theoretic structures, whose computation rules are given by typal equalities. Our main result is that any model of type theory with weakly stable weak type-theoretic structures admits an equivalent model with strictly stable weak type-theoretic structures.


page 1

page 3

page 7

page 19


Globular Multicategories with Homomorphism Types

We introduce a notion of globular multicategory with homomorphism types....

Type-theoretic weak factorization systems

This article presents three characterizations of the weak factorization ...

The Metric Space of Networks

We study the question of reconstructing a weighted, directed network up ...

Multi-stable design of triangulated origami structures on cones of revolution

It is well-known that the Kresling pattern of congruent triangles can be...

Approximately Stable Committee Selection

In the committee selection problem, we are given m candidates, and n vot...

Tietze Equivalences as Weak Equivalences

A given monoid usually admits many presentations by generators and relat...

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types

How do spaces emerge from pregeometric discrete building blocks governed...