[Paper Review] A Stratified Approach to Löb Induction

Daniel Gratzer, Michael Shulman|arXiv (Cornell University)|Jan 1, 2022

Computability, Logic, AI Algorithms3 citations

TL;DR

This paper presents a stratified construction of cumulative universes in Grothendieck topoi that satisfy the realignment property—a key condition for modeling Martin-Löf type theory with cumulative universes and for synthetic Tait computability. By adapting a small object argument and leveraging descent theory, the authors lift a well-behaved universe hierarchy from the category of sets to any Grothendieck topos, ensuring coherence across all universe levels and enabling direct interpretations of type theory in all such topoi.

ABSTRACT

Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type theories remains underdeveloped. We show that integrating Löb induction is the key obstruction to unifying guarded recursion and dependence in a well-behaved type theory and prove a no-go theorem sharply bounding such type theories. Based on these results, we introduce GuTT: a stratified guarded type theory. GuTT is properly two type theories, sGuTT and dGuTT. The former contains only propositional rules governing Löb induction but enjoys decidable type-checking while the latter extends the former with definitional equalities. Accordingly, dGuTT does not have decidable type-checking. We prove, however, a novel guarded canonicity theorem for dGuTT, showing that programs in dGuTT can be run. These two type theories work in concert, with users writing programs in sGuTT and running them in dGuTT.

Motivation & Objective

• To resolve the failure of sheafification to preserve the realignment property critical for univalent type theory and synthetic metatheory.
• To extend Hofmann and Streicher's presheaf universe construction to all Grothendieck topoi while preserving cumulativity and realignment.
• To provide a direct interpretation of Martin-Löf type theory with cumulative universes in any Grothendieck topos.
• To support the application of synthetic methods—especially Artin gluing and Tait computability—to all Grothendieck topoi.
• To lay the foundation for a constructive version of the construction, though this remains open.

Proposed method

• Adapts Shulman’s universe construction via a small object argument based on κ-compactness and saturation of realignment problems.
• Uses descent theory in Grothendieck topoi to ensure that the universe structure lifts coherently across the topos.
• Employs the adjunction j! ⊣ j* for a closed subtopos j: F → G to construct cartesian lifts of morphisms, ensuring realignment.
• Applies Frobenius reciprocity and the strictness of the initial object to verify that key diagrams (e.g., Diagram 47) are cartesian.
• Constructs a generic family for the universe by combining internal presheaf constructions with gluing techniques.
• Verifies that the resulting universe satisfies all axioms (U1–U8), including the critical realignment condition (U8), via diagrammatic and adjoint reasoning.

Experimental results

Research questions

• RQ1 Can a cumulative universe hierarchy satisfying realignment be constructed in any Grothendieck topos, even after sheafification?
• RQ2 Does the realignment property—essential for univalent type theory and synthetic Tait computability—survive in sheaf topoi when using standard sheafification?
• RQ3 Can a universe construction in a Grothendieck topos be made fully coherent across all universe levels while preserving cumulativity?
• RQ4 Is there a constructive version of the universe construction that avoids reliance on choice and classical logic?
• RQ5 Can the realignment property be uniformly lifted across all monomorphisms in a topos without choice?

Key findings

• A cumulative universe hierarchy satisfying the realignment property (U8) is constructed in any Grothendieck topos, using a stratified small object argument.
• The construction ensures that the universe satisfies all axioms (U1–U8), including the critical realignment condition, even in sheaf topoi.
• The realignment property is preserved under base change via j* for any closed subtopos j: F → G, enabling the repair of failed lifts via Diagram 47.
• The construction extends the Hofmann–Streicher interpretation of Martin-Löf type theory with cumulative universes to all Grothendieck topoi.
• The method supports applications in synthetic Tait computability and the semantics of cubical type theory by ensuring strict preservation of codes under j*.
• A constructive version of the construction remains open, though the authors show that (U8) holds for decidable monomorphisms in the absence of choice.