Strict universes for Grothendieck topoi

TL;DR

This paper constructs cumulative universe hierarchies with the realignment property in all Grothendieck topoi, enabling direct interpretations of Martin-Löf type theory and extending synthetic methods in type theory semantics.

Contribution

It introduces a slight modification of Shulman's argument to build cumulative universes with the realignment property in any Grothendieck topos, broadening the applicability of synthetic methods.

Findings

Cumulative universe hierarchies with realignment property are constructed in all Grothendieck topoi.

Direct-style interpretations of Martin-Löf type theory with cumulative universes are possible.

Extension of synthetic methods in type theory semantics to all Grothendieck topoi.

Abstract

Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In parallel, van den Berg and Moerdijk have shown in the context of algebraic set theory that similar constructions continue to apply even in weaker metatheories. Unfortunately, sheafification seems not to preserve an important realignment property enjoyed by the presheaf universes that plays a critical role in models of univalent type theory as well as synthetic Tait computability, a recent technique to establish syntactic properties of type theories and programming languages. In the context of multiple universes, the realignment property also implies a coherent choice of codes for connectives at each universe level, thereby interpreting the cumulativity laws present in popular formulations of Martin-L\"of type theory. We observe that a slight adjustment to an argument of Shulman constructs a cumulative universe hierarchy satisfying the realignment property at every level in any Grothendieck topos. Hence one has direct-style interpretations of Martin-L\"of type theory with cumulative universes into all Grothendieck topoi. A further implication is to extend the reach of recent synthetic methods in the semantics of cubical type theory and the syntactic metatheory of type theory and programming languages to all Grothendieck topoi.