[Paper Review] Formalising Inductive and Coinductive Containers

Damato, Stefania, Altenkirch, Thorsten+1 more|arXiv (Cornell University)|Feb 22, 2014

Logic, programming, and type systems3 references17 citations

TL;DR

This paper introduces univalent models of Martin-Löf type theory by constructing contextual categories from Kan fibrations in the category of simplicial sets, using a universal ordered Kan fibration p<α with fibers of cardinality less than a large cardinal α. The key contribution is proving that these models satisfy the univalence axiom—equivalent types are equal—thereby providing a semantics that supports a formalization of mathematics where isomorphic structures are identified, with the space of such structures being contractible, ensuring uniqueness up to homotopy.

ABSTRACT

Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of locally cartesian closed categories (LCCCs) with disjoint coproducts and W-types, and uniqueness of identity proofs (UIP) is implicitly assumed throughout. Although it is claimed that these developments can also be interpreted in extensional Martin-Löf type theory, this interpretation is not made explicit. In this paper, we present a formalisation of the results that "containers preserve least and greatest fixed points" in Cubical Agda, thereby giving a formulation in intensional type theory. Our proofs do not make use of UIP and thereby generalise the original results from talking about container functors on Set to container functors on the wild category of types. Our main incentive for using Cubical Agda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of Cubical Agda’s path type to coinductive proofs.

Motivation & Objective

• To establish a new class of models for Martin-Löf type theory that satisfy the univalence axiom.
• To formalize the semantics of type theory using contextual categories derived from universal fibrations in simplicial sets.
• To demonstrate that these models are consistent with ZFC and support a homotopically coherent formalization of mathematics.
• To provide a systematic framework linking type systems to initial models of quasi-equational theories.
• To lay the foundation for a formal, automated verification of type-theoretic systems via quasi-equational specifications.

Proposed method

• Constructs contextual categories CC(C, p) from a locally cartesian closed category C and a universe p: eU → U with pull-backs.
• Uses the category of simplicial sets ∆opSets as a foundational model, leveraging its structure as a Grothendieck topos and locally cartesian closed category.
• Defines a universal ordered morphism eV<α → V<α with fibers of size < α, from which a universal Kan fibration p<α: eU<α → U<α is extracted.
• Proves that U<α is a Kan simplicial set (Theorem 5), ensuring good homotopical behavior.
• Shows that p<α is univalent (Theorem 7), meaning the morphism mq: B → Eq(E×B, B×E) is a weak equivalence, which encodes the univalence principle.
• Demonstrates that the space of structures (products, sums, equality, universes) on p<α is contractible (Theorem 3), implying essential uniqueness.

Experimental results

Research questions

• RQ1 Can a model of Martin-Löf type theory be constructed that satisfies the univalence axiom and is consistent with ZFC?
• RQ2 How can type systems be formally connected to initial models of quasi-equational theories?
• RQ3 What role do Kan fibrations in simplicial sets play in realizing homotopical semantics for type theory?
• RQ4 Is the space of structures (e.g., dependent products, equality) on a universal fibration p<α essentially unique?
• RQ5 Can univalent models be used to formalize mathematics in a way that identifies isomorphic structures?

Key findings

• The contextual category CC(∆opSets, p<α) forms a univalent model of type theory, satisfying the univalence axiom.
• The space of structures (products, sums, equality, universes) on p<α is contractible, ensuring that such structures are unique up to homotopy.
• The fibration p<α is univalent, which directly encodes the principle that equivalent types are equal.
• The construction provides a model of type theory that satisfies both the univalence axiom and the law of excluded middle, proving consistency relative to ZFC.
• The initial model of the quasi-equational theory corresponding to type systems can be realized in ZFC via this construction.
• The framework supports a formal, systematic, and potentially automatable link between type systems and their semantic models.