[Paper Review] An inductive-recursive universe generic for small families

Daniel Gratzer|arXiv (Cornell University)|Feb 11, 2022

Advanced Topology and Set Theory4 citations

TL;DR

This paper presents a generic inductive-recursive universe construction in Grothendieck topoi that supports observational type theory (TT obs) with injective type constructors. By using a small inductive-recursive definition with a strict Tarski-style universe and a lifting operation, the authors ensure cumulativity and genericity for small families, enabling TT obs as a viable internal language across diverse categorical models.

ABSTRACT

We show that it is possible to construct a universe in all Grothendieck topoi with injective codes a la Pujet and Tabareau which is nonetheless generic for small families. As a trivial consequence, we show that their observational type theory admits interpretations in Grothendieck topoi suitable for use as internal languages.

Motivation & Objective

• To resolve the semantic incompatibility of injective type constructors in standard set-theoretic models of observational type theory (TT obs).
• To extend TT obs to arbitrary Grothendieck topoi by constructing a universe that remains generic for small families.
• To demonstrate that a small inductive-recursive universe with a strict Tarski-style encoding supports cumulativity and internal type theory in complex categorical settings.
• To show that the standard inductive-recursive construction can be adapted to preserve genericity for relatively κ-compact families.

Proposed method

• Uses a small inductive-recursive definition to construct a universe V_i in a Grothendieck topos, simultaneously defining a type family A:U_1 and a decoding function r:El_V_i(A) → type.
• Introduces a lifting operation 'up' that maps types A to codes in a higher universe, ensuring strict cumulativity across universe levels.
• Employs a Tarski-style universe with explicit decoding function El, ensuring that type constructors like Π and Σ commute with codes up to isomorphism.
• Modifies the standard inductive-recursive universe to ensure that the universe remains generic for small families, including relatively κ-compact families.
• Applies the construction in a generalized algebraic theory framework, abstracting from coherence and partial interpretation issues.
• Uses the 'up' construction to embed lower-level types into higher universes, preserving type structure and enabling strict hierarchy.

Experimental results

Research questions

• RQ1 Can an inductive-recursive universe be constructed in Grothendieck topoi that supports injective type constructors required by TT obs?
• RQ2 How can a universe be made generic for small families while maintaining cumulativity in categorical models?
• RQ3 Is it possible to achieve a strictly cumulative hierarchy in TT obs using small induction-recursion and Tarski-style universes?
• RQ4 Can the resulting universe construction serve as a viable internal language in arbitrary Grothendieck topoi?
• RQ5 What modifications to standard inductive-recursive universes ensure genericity for families beyond the standard universe hierarchy?

Key findings

• A small inductive-recursive universe can be constructed in any Grothendieck topos that supports injective type constructors, resolving semantic issues in standard set-theoretic models.
• The construction ensures that the universe remains generic for small families, including relatively κ-compact families, which is essential for internal type theory.
• The use of a strict Tarski-style universe with the 'up' lifting operation enables a strictly cumulative hierarchy that supports decidable type-checking.
• The model satisfies all required type constructor isomorphisms, ensuring that Π and Σ types are correctly interpreted via code decoding.
• The approach allows TT obs to be interpreted in arbitrary Grothendieck topoi, establishing it as a viable internal language in diverse categorical settings.
• The method generalizes the construction from [PT22] to all Grothendieck topoi, extending its applicability beyond setoid models.