Towards coherence theorems for equational extensions of type theories

TL;DR

This paper investigates the conservativity of extending dependent type theories with additional equalities, aiming to generalize known results to theories like Homotopy Type Theory by constructing an infinity-congruence and analyzing its properties.

Contribution

It introduces a method to analyze equational extensions of type theories via infinity-congruences, generalizing conservativity results beyond extensional theories.

Findings

Constructed an infinity-congruence for base theories with added equations.

Proved the second factor of the extension is an equivalence when the infinity-congruence is 0-truncated.

Conjectured the first factor is always an equivalence for well-behaved base theories.

Abstract

We study the conservativity of extensions by additional strict equalities of dependent type theories (and more general second-order generalized algebraic theories). The conservativity of Extensional Type Theory over Intensional Type Theory was proven by Hofmann. Our goal is to generalize such results to type theories without the Uniqueness of Identity Proofs principles, such as variants of Homotopy Type Theory. For this purpose, we construct what is essentially the infinity-congruence on the base theory that is freely generated by the considered equations. This induces a factorization of any equational extension, whose two factors can be studied independently. We conjecture that the first factor is always an equivalence when the base theory is well-behaved. We prove that the second factor is an equivalence when the infinity-congruence is 0-truncated.