Sheaves, Cosheaves and Applications (1303.3255v2)

Abstract: This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.

An Exploration of Sheaves and Cosheaves in Mathematical and Applied Contexts

The paper under consideration delivers a detailed exposition of sheaves and cosheaves, focusing on their mathematical underpinnings as well as elaborating on their applicability in various domains of science and engineering. This document comprises several parts, each addressing a unique aspect of sheaf theory and its interactions with other mathematical constructs. The author structures the document to cater to multiple audience segments, ranging from neophytes to advanced researchers, providing a comprehensive introduction to the core concepts and extending to more complex theoretical advancements.

The initial part of the document establishes foundational knowledge in the field of category theory, sheaves, and cosheaves. For researchers not deeply acquainted with category theory, guidance is provided to navigate the initial chapters, focusing on the essential elements before engaging with the more intricate examples and abstractions of sheaves on topological spaces. Significant attention is given to the cohesiveness of sheaf and cosheaf definitions, promoting a clearer understanding by leveraging explicit examples and demystifying complex concepts such as the sheaf axiom and \v{C}ech homology.

Progressing beyond the basics, the second part reveals the application of cellular sheaves and cosheaves through the lens of linear algebra over cell complexes. The manuscript embarks on justifying the usage of the term “sheaf” by linking it with Alexandrov topology, which transforms posets into sheaves or cosheaves. The authors also incorporate the notion of “barcodes” as a novel methodological tool for interpreting cellular sheaf cohomology and cosheaf homology. This part vigorously develops the computational framework for cellular sheaf cohomology and cosheaf homology, culminating in their integration with derived categories. These contributions are noteworthy in that they not only operationalize complex mathematical constructs but also present a pathway for practical computation.

The third section details the utility of cellular sheaves and cosheaves in scientific and engineering scenarios. Chapter discussions emphasize the reformulation of persistent homology using sheaves and cosheaves, improving homology computations' distributive and aggregative properties. This reformulation is substantiated through a new theorem that connects level set and sub-level set persistence via spectral sequences. In a foray into applied fields, the paper examines how sheaves contribute to network coding and sensor networks, shedding light on the computational advantages and theoretical insights sheaves provide in these domains.

In its final part, the paper explores the intricate mathematical contributions making up the core of the thesis. This entails thorough explorations into the equivalences of constructible cosheaves and representations of MacPherson's entrance path category, exemplifying advanced applications of stratification theory. The discussion extends to proving a codimension criterion for Thom’s condition $a_f$, establishing a connection with Verdier duality, and articulating duality in terms of derived equivalences. Additionally, it introduces the concept of an interleaving distance for sheaves on a metric space, illustrating its fundamental metric properties.

Overall, the paper significantly contributes to the field of algebraic topology and its interfaces with applied mathematics by presenting robust theoretical breakthroughs and their corresponding computational and practical frameworks. The research holds implications for future developments in AI and computational topology, offering tools and insights with broad applicability. Future explorations may further probe into optimizing the computational aspects or exploring additional applications in multi-dimensional data analysis and beyond.