On the Local Kan Structure and Differentiation of Simplicial Manifolds

Core Concepts

Simplicial manifolds possess a local Kan structure that allows for their differentiation into higher Lie algebroids, extending the differentiation methods from higher Lie groupoids to a broader class of simplicial structures.

Abstract

• Bibliographic Information: Dorsch, F. (2024). On the local Kan structure and differentiation of simplicial manifolds. arXiv:2402.07857v2 [math.DG].

• Research Objective: This paper investigates the local structure of simplicial manifolds and aims to establish a method for their differentiation, extending the existing techniques for higher Lie groupoids.

• Methodology: The author leverages the concept of Kan conditions, which are horn-filling conditions in simplicial objects, to analyze the local behavior of simplicial manifolds. By demonstrating that simplicial manifolds satisfy Kan conditions locally, the author paves the way for defining a tangent complex and a differentiation functor analogous to those used for higher Lie groupoids.

• Key Findings: The paper's central finding is that simplicial manifolds inherently possess a local Kan structure. This means that while they may not satisfy Kan conditions globally, they do so in a neighborhood of their base space. This local structure is sufficient to define a tangent complex for simplicial manifolds, which captures their infinitesimal behavior. Furthermore, the author proves that the tangent functor, which assigns a simplicial manifold to a presheaf of graded manifolds, is representable by this tangent complex.

• Main Conclusions: The existence of a well-defined tangent complex and the representability of the tangent functor provide a concrete method for differentiating simplicial manifolds into higher Lie algebroids. This extends the existing differentiation techniques from the realm of higher Lie groupoids to a broader class of simplicial structures.

• Significance: This research significantly contributes to the field of higher structures by providing a new perspective on the differentiation of simplicial manifolds. It offers a powerful tool for studying the infinitesimal structure of these objects, which are prevalent in various areas of mathematics and theoretical physics.

• Limitations and Future Research: The paper primarily focuses on establishing the theoretical framework for differentiating simplicial manifolds. Further research is needed to explore the computational aspects of this method and to investigate its applications in specific examples of simplicial manifolds arising from different geometric and topological contexts.

Stats

Quotes

"Unlike simplicial sets and simplicial schemes, for which one can readily find many examples which fail to satisfy Kan conditions, it is hard to find natural and interesting examples of simplicial manifolds that do not satisfy (at least weak versions of) Kan conditions."
"The project initiated in [LRWZ23] and set to be concluded in a subsequent article aims to show that any higher Lie groupoid X can be differentiated to a higher Lie algebroid whose underlying graded vector bundle is described by the tangent complex of X."
"Many naturally occurring objects, such as those derived from singular foliations, often possess inherent simplicial structures."

Key Insights Distilled From

On the local Kan structure and differentiation of simplicial manifolds

by Florian Dors... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2402.07857.pdf

Deeper Inquiries

How does the local Kan structure of simplicial manifolds relate to their global topological and geometric properties?

The local Kan structure of simplicial manifolds, as established in the paper, provides a powerful lens through which we can examine the interplay between their local and global characteristics. Here's a breakdown of this relationship:

Bridging Local and Global: The local Kan structure, essentially stating that simplicial manifolds behave like Kan complexes in small neighborhoods of their base space, allows us to transfer tools and insights from the well-studied realm of Kan complexes to the more intricate setting of simplicial manifolds. This bridge is crucial because while global properties might be complex and difficult to grasp directly, local properties are often more tractable.

Tangent Complex as a Global Invariant: The paper leverages the local Kan structure to define the tangent complex of a simplicial manifold. Despite being defined locally, the tangent complex captures crucial global information about the simplicial manifold. It serves as a global invariant, encoding data about the infinitesimal structure and potential 'directions' of movement within the simplicial manifold.

Obstructions to Global Kan Conditions: The existence of a local Kan structure doesn't necessarily imply that the simplicial manifold is a Kan complex globally. Global obstructions to satisfying the Kan conditions can arise from the global topology of the manifold. The local Kan structure, however, allows us to study these obstructions by analyzing how the local Kan conditions break down as we move away from the base space.

Relationship with Geometric Structures: The tangent complex, derived from the local Kan structure, can be further equipped with additional geometric structures like Lie algebroids or L-infinity algebras. These structures provide a powerful framework for studying geometric notions like curvature, holonomy, and characteristic classes in the context of simplicial manifolds.
In essence, the local Kan structure provides a crucial link between the local and global aspects of simplicial manifolds. It allows us to define global invariants like the tangent complex and provides a framework for studying how local geometric structures piece together to influence the global properties of these objects.

Could there be alternative approaches to differentiating simplicial manifolds that do not rely on the Kan structure, and if so, how would they compare to the method presented in this paper?

Yes, alternative approaches to differentiating simplicial manifolds that don't explicitly rely on the Kan structure are conceivable. Here are a few possibilities and their comparison to the Kan-based method:

Direct Geometric Differentiation:

Idea: Instead of relying on the combinatorial structure of horns and fillers, one could attempt to directly differentiate the structure maps of the simplicial manifold. This might involve considering tangent maps of face and degeneracy maps and studying their interplay.
Comparison: This approach is more geometrically intuitive but could be significantly more technically challenging. Establishing the necessary smoothness properties and compatibility conditions for the differentiated structure maps might be difficult without leveraging the local Kan structure.

Sheaf-Theoretic Techniques:

Idea: Simplicial manifolds can be viewed as representable sheaves on an appropriate site. One could explore differentiating these sheaves directly using tools from sheaf theory and derived geometry.
Comparison: This approach is very abstract and would likely require a significant amount of technical machinery. However, it could potentially provide a more general framework for differentiation, applicable to a broader class of objects beyond simplicial manifolds.

Simplicial Homotopy Theory:

Idea: Simplicial manifolds naturally live in the world of simplicial sets, where homotopy-theoretic notions are central. One could explore differentiating simplicial manifolds by considering their tangent complexes within a suitable model category of simplicial sets.
Comparison: This approach aligns well with the homotopy-theoretic nature of simplicial objects but might obscure the geometric intuition behind differentiation. It could be particularly useful for studying homotopy invariants of simplicial manifolds.

Comparison to the Kan-Based Method:
The Kan-based method presented in the paper has the advantage of being relatively concrete and computationally tractable. The local Kan structure provides a clear roadmap for constructing the tangent complex and relating it to the underlying geometry of the simplicial manifold. However, alternative approaches might offer greater generality or deeper connections to other areas of mathematics.

What are the implications of this research for understanding the relationship between discrete and continuous structures in mathematics and physics?

This research on differentiating simplicial manifolds has profound implications for understanding the intricate relationship between discrete and continuous structures, a theme that permeates various areas of mathematics and physics:

Unifying Discrete and Continuous: Simplicial manifolds themselves embody a fusion of discrete (simplicial structure) and continuous (manifold structure) aspects. The ability to differentiate them provides a concrete way to transition from the discrete to the continuous, capturing infinitesimal information encoded in the discrete data.

Quantization and Discretization: In physics, quantization often involves replacing continuous spaces with discrete approximations. This research could offer insights into the reverse process – how to recover continuous structures from discrete or quantized models. The tangent complex, derived from the discrete simplicial data, could provide a way to reconstruct a 'classical' continuous limit.

Discrete Approximations in Geometry and Topology: Many geometric and topological invariants are defined for continuous objects. The ability to differentiate simplicial manifolds could lead to new ways of approximating these invariants using discrete combinatorial data. This has implications for computational topology and geometry, where discrete representations are often necessary.

Applications in Network Theory and Data Analysis: Simplicial complexes are increasingly used to model complex networks and data sets. The differentiation of simplicial manifolds could provide tools for studying the 'dynamics' and evolution of these networks, capturing how infinitesimal changes in the network structure affect its global properties.

Bridging Different Mathematical Fields: This research builds bridges between seemingly disparate areas like differential geometry, algebraic topology, and category theory. This cross-fertilization of ideas can lead to new perspectives and breakthroughs in each of these fields.
In conclusion, the ability to differentiate simplicial manifolds provides a powerful new tool for studying the interplay between discrete and continuous structures. This has far-reaching implications for various areas of mathematics and physics, potentially leading to new insights into quantization, discretization, geometric invariants, and the dynamics of complex systems.