Homotopy Type Theory: Univalent Foundations of Mathematics (1308.0729v1)

Abstract: Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.

• The paper introduces a homotopical interpretation of type theory, redefining types as structured spaces that admit paths as identifications.
• It demonstrates that identity types function as path spaces, allowing robust reasoning about equivalences beyond classical equality.
• The work advances univalent foundations and higher inductive types, paving the way for computer-assisted proofs and unified mathematical discourse.

An Overview of "Homotopy Type Theory: Univalent Foundations of Mathematics"

The manuscript "Homotopy Type Theory: Univalent Foundations of Mathematics" introduces transformative concepts in mathematical foundations, presenting a new paradigm where type theory is interpreted through the lens of homotopy and higher groupoid structures. This innovative approach stems from the Univalent Foundations Program and seeks to unify and extend classical mathematics into a framework that integrates type theory with insights from homotopy theory and higher category theory.

Key Concepts and Contributions

1. Interpretation of Types as Spaces: Types in Homotopy Type Theory (HoTT) are not merely seen as collections of elements but are imbued with the structure of spaces, akin to topological spaces in classical homotopy theory or objects in higher category theory. This allows types to support paths, or homotopies, between elements, representing identifications or equivalences rather than strict equalities.
2. Identity Types as Path Spaces: Identity types represent paths between elements, capturing the homotopical notion of deformation, rather than mere propositional equality. This provides a rich structure for reasoning about equivalence beyond traditional logical paradigms.
3. Function Extensionality and Univalence Axiom: The Univalence Axiom, a central component of HoTT, posits that equivalent types can be identified — effectively allowing the substitution of isomorphic structures across mathematical discourse. Function extensionality further ensures that functions equal at every input are themselves equal, supporting a more flexible notion of function equality pertinent for both theoretical and practical applications in proof assistants.
4. Higher Inductive Types: HoTT extends the concept of types to include Higher Inductive Types (HITs), allowing the construction of new types along with specified higher-dimensional paths. This provides a robust framework for defining spaces like spheres or more complex objects entirely within the type-theoretic system, aligning closely with practices in algebraic topology.
5. Consequences for Logic: The manuscript explores the implications of its homotopical viewpoint on traditional logical principles. It challenges classical logic, particularly regarding the law of excluded middle and the axiom of choice, reflecting its foundational stance in constructive mathematics.

Implications and Future Directions

Practically, HoTT offers the promise of a seamless integration with computer-assisted proof systems, democratizing access to rigorous mathematical verification and exploration. Theoretically, it posits a global framework that harmonizes disparate mathematical domains through its unified language of types and equivalences, while providing new tools for tackling complex mathematical problems like the computation of homotopy groups of spheres.

The potential future developments include further refinement and expansion of Higher Inductive Types, a deeper exploration of the implications of univalence in specific mathematical contexts, and the advancement of software systems that leverage HoTT for interactive theorem proving. Additionally, continued research is needed to fully realize the implications of HoTT on existing mathematical structures and to explore its capacity to generate new mathematical insights.

Conclusion

"Homotopy Type Theory: Univalent Foundations of Mathematics" stands as a pivotal work, challenging foundational assumptions and proposing a comprehensive, integrated framework that reshapes how mathematicians and computer scientists approach type theory and homotopy. By marrying the rigor of formal methods with the conceptual agility of homotopy, HoTT sets the stage for transformative advances across theoretical and applied mathematics.