Sheaves, Cosheaves and Applications

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.

v2: 307 pages, 68 figures, includes minor edits of version 1 and is expanded with additional results, accepted as a doctoral thesis at the University of Pennsylvania. v1: 188 pages, 47 figures, work in progress

References in corpus (5)

• On Derived Equivalences of Categories of Sheaves Over Finite Posets
• Dynamics on Networks of Manifolds
• Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture
• Derived Algebraic Geometry VI: E_k Algebras
• Popaths and Holinks

Cited by in corpus (45)

• A roadmap for the computation of persistent homology
• Metrics for generalized persistence modules
• Interactive Visualization of 2-D Persistence Modules
• Generalized Persistence Diagrams
• Algebraic Stability of Zigzag Persistence Modules
• Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories
• The ${\ell }^{\infty }$-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance
• Ephemeral persistence modules and distance comparison
• Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
• Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem
• Data structures for real multiparameter persistence modules
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing
• Local cohomology and stratification
• Assignments to sheaves of pseudometric spaces
• Lefschetz section theorems for tropical hypersurfaces
• Discrete Morse theory and localization
• Weyl n-algebras
• Functors on Posets Left Kan Extend to Cosheaves: an Erratum
• Persistent homology and microlocal sheaf theory
• Locally standard torus actions and sheaves over Buchsbaum posets
• Understanding networks and their behaviors using sheaf theory
• A family of metrics from the truncated smoothing of Reeb graphs
• Essential graded algebra over polynomial rings with real exponents
• Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives
• Maximal Sections of Sheaves of Data over an Abstract Simplicial Complex
• Homology of torus spaces with acyclic proper faces of the orbit space
• Extra-fine sheaves and interaction decompositions
• Topos and Stacks of Deep Neural Networks
• Modules over posets: commutative and homological algebra
• Algebra of the infrared and secondary polytopes
• Quiver Signal Processing (QSP)
• Weighted Persistent Homology
• Combinatorial Persistent Homology Transform
• Cosheaf Theoretical Constructions in Networks and Persistent Homology
• Harder-Narasimhan Filtrations of Persistence Modules
• Sheaf Neural Networks
• Path Optimization Sheaves
• Quantitative Tamarkin category
• Reeb Graph Metrics from the Ground Up
• Discrete Morse theory for computing cellular sheaf cohomology
• Constraints on pure point diffraction on aperiodic point patterns of finite local complexity
• Piecewise linear sheaves
• Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation
• Blow up of conductors
• Canonical stratifications along bisheaves