Discrete differential geometry in homotopy type theory (2504.20363v1)

Abstract: Type families on higher inductive types such as pushouts can capture homotopical properties of differential geometric constructions including connections, curvature, and vector fields. We define a class of pushouts based on simplicial complexes, then define principal bundles, connections, and curvature on these. We provide an example of a tangent bundle but do not prove when these must exist. We define vector fields, and the index of a vector field. Our main result is a theorem relating total curvature and total index, a key step to proving the Gauss-Bonnet theorem and the Poincar\'e-Hopf theorem, but without an existing definition of Euler characteristic to compare them to. We draw inspiration in part from the young field of discrete differential geometry, and in part from the original classical proofs, which often make use of triangulations and other discrete arguments.

• The paper introduces a framework for discrete differential geometry within homotopy type theory, realizing constructs like connections and curvature using higher inductive types on simplicial complexes.
• It applies this framework to analyze theorems such as Gauss-Bonnet and Poincaré-Hopf, offering novel index calculations without relying on the Euler characteristic.
• This work provides a foundation for formalizing topology and gauge theory in HoTT and suggests potential for verifying geometric computations and exploring classic invariants.

Discrete Differential Geometry in Homotopy Type Theory

This work explores discrete differential geometry through the lens of homotopy type theory (HoTT), presenting a framework for combining differential geometry and the type-theoretic approach to homotopy. The paper focuses on the realization of differential geometric constructs such as connections, curvature, and vector fields using higher inductive types, specifically pushouts based on simplicial complexes.

Key Contributions and Methods

The paper introduces methods to define principal bundles, connections, and curvature on simplicial complexes and evaluates the homotopical properties of these constructs. The discrete approach provided in HoTT allows examination of classical theorems, notably the Gauss-Bonnet and Poincaré-Hopf theorems. Unlike classical methods, the authors do not rely on the Euler characteristic, creating a novel way to analyze the relationships and implications of these theorems. The authors define vector fields using tangent bundles without providing a comprehensive proof of their existence, highlighting the potential challenges and areas for future research.

Technical Details and Results

1. Simplicial Complex Realization: A sequence of pushouts is used to realize simplicial complexes as higher inductive types. These are structured stepwise through dimensions, from the 0-dimensional complex to higher-dimensional complexes, emphasizing the complex's cellular structure and its role in differential geometry.
2. Principal Bundles and Fibrations: Utilizing Scoccola's results, the paper outlines when maps factor through principal fibrations, connecting higher inductive types to classical topology.
3. Vector Fields and Index Calculation: The paper defines vector fields as sections of tangent bundles over 2-dimensional cellular types, introducing swirling and cancellation methods to compute vector field indices with respect to orientable simplicial complexes.
4. Octahedron Model of the Sphere: An example of a combinatorial sphere using an octahedron is presented. This realization is leveraged to explore the tangent bundle and extend vector fields, illustrating how geometric data can model differential structures within HoTT.

Implications and Future Directions

The framework established provides insights into incorporating differential geometry constructs in type theory, potentially opening new pathways for formalizing topology and gauge theory in HoTT. This suggests broader applications, such as using these models for validating geometric computations directly within a type-theoretical setting. The paper posits regions for extending the results, such as proving the factorization of bundles through Eilenberg-Mac Lane spaces and deriving internal definitions of classic invariants like the Euler characteristic. Furthermore, the work lays a foundation for classifying spaces and studying Chern-Weil theory, aiming to develop homotopical results accessible in HoTT.

To advance this field, further investigation into the integration of discrete differential geometry principles and HoTT is necessary. This could involve formalizing the presented examples, extending to higher dimensions, and assessing how existing discrete approaches align with or enhance classical differential geometric methods. Ultimately, the richness of the type-theoretic approach could lead to substantial advancements in understanding and applying differential geometry across mathematical disciplines.