Cubical Type Theory: a constructive interpretation of the univalence axiom (1611.02108v1)

Abstract: This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

• The paper achieves a constructive proof of the univalence axiom by developing a cubical set model that manipulates n-dimensional cubes.
• The paper streamlines identity and transport operations by introducing composition techniques that simplify handling of path types and function extensionality.
• The paper extends type theory with higher inductive types and glueing constructions, enhancing both theoretical insights and practical applications in proof assistants.

Overview of "Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom"

The paper "Cubical Type Theory: A Constructive Interpretation of the Univalence Axiom" by Cyril Cohen, Thierry Coquand, Simon Huber, and Anders Mörtberg presents a novel type theoretic framework that enables direct manipulation of $n$-dimensional cubes within a cubical set model, leading to a constructive interpretation of Homotopy Type Theory (HoTT). It achieves this interpretation with the inclusion of the univalence axiom, advanced handling of identity types, and the introduction of higher inductive types (HITs).

Core Contributions

1. Cubical Set Model: The paper develops a type theory based on cubes, where geometric intuitions are represented through types. This approach allows for direct manipulation of identity types and provides a means to handle complex operations like function extensionality and path types without relying on traditional syntactic equality.
2. Univalence Axiom: A critical achievement of this framework is the constructive proof of the univalence axiom, which equates isomorphic structures. This is accomplished by proving that the operation of moving between types along paths parallels the equivalence between these types. The authors utilize a geometric interpretation via higher-dimensional cubes to achieve this, differing from previous semantic models based on simplicial sets.
3. Composition and Transport: The inclusion of a composition operation for paths and types allows for a rich handling of equality. The concept of transport in type theory, usually a rather complex operation, is streamlined in this model using direct geometric analogues, enhancing the computational capabilities.
4. Higher Inductive Types (HITs): The paper extends the basic type system to include higher inductive types such as spheres and propositional truncation. These HITs are incorporated seamlessly into the cubical framework, providing new syntactic elements that enhance reasoning within the type theory.
5. Contractibility and Glueing: The framework defines advanced notions such as contractible types and a glueing construction that maintains consistencies in types through equivalence structures. These notions are pivotal for constructing proofs and transformations within the type theory, enhancing its logical foundation.

Implications and Future Directions

The introduction of cubical type theory offers significant advancements both theoretically and practically. This framework not only serves as an alternative way to realize the principles of Homotopy Type Theory but also provides practicality in mechanizing mathematics using proof assistants.

• Theoretical Implications:

The work enhances our understanding of the interplay between type theory and homotopy theory, deepening the correspondence between topological spaces and types. The successful constructive proof of univalence affirms the potential of cubical approaches to explore new directions in type-theoretic foundations.

• Practical Implications:

As this framework matures, it holds promise for impacting tools like proof assistants, enabling them to directly use homotopic principles more effectively and efficiently. Implementation like the cubicaltt highlights the readiness of these ideas for practical computation.

• Future Research:

The paper outlines several avenues for further exploration: extending the semantics to incorporate all inductive families, generating a broader syntax for HITs, and examining resizing rules. These directions emphasize the ongoing development and scalability of cubical type theory.

The authors achieve an intricate balance between theory and application, offering a robust foundation for exploiting higher-dimensional and homotopic thinking in type theory. Overall, this work solidifies a significant step in formalizing and expanding the utility of Homotopy Type Theory through a cubical lens.