Introduces game sheaves in a Grothendieck topos of time-space histories where Nash equilibria appear as global sections of a best-response correspondence sheaf.
Sheaves, Cosheaves and Applications
6 Pith papers cite this work. Polarity classification is still indexing.
6
Pith papers citing it
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.
verdicts
UNVERDICTED 6representative citing papers
Sheaf-ADMM trains multi-agent systems by unrolling ADMM with sheaf-specified constraints, yielding improved MNIST robustness to shifts and higher Sudoku solve rates than MPNN baselines.
Liability clearing in networks is modeled as global sections of a liability sheaf on directed hypergraphs, identified as a finite-limit construction with existence and uniqueness from lattice and metric theorems on payment objects.
A finite sheaf-theoretic framework ranks obstruction measures to identify when an AI agent's theory must deform within its language or extend to a new one, validated on a controlled transition benchmark.
HetSheaf applies cellular sheaves and type-conditioned restriction maps to heterogeneous graphs, plus SheafPool for basis-invariant graph-level representations, delivering competitive accuracy with substantially reduced parameter counts.
Proposes a geometric-categorical synthesis using Riemannian flows, Clifford rotors, sheaf Laplacians, and topos logic for resilient multi-agent consensus and planning.
citing papers explorer
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A Sheaf Framework for Strategic Multi-Agent Systems: From Consensus to Nash Equilibria
Introduces game sheaves in a Grothendieck topos of time-space histories where Nash equilibria appear as global sections of a best-response correspondence sheaf.
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Learning Multi-Agent Coordination via Sheaf-ADMM
Sheaf-ADMM trains multi-agent systems by unrolling ADMM with sheaf-specified constraints, yielding improved MNIST robustness to shifts and higher Sudoku solve rates than MPNN baselines.
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Clearing in Liability Networks via Sheaves on Directed Hypergraphs
Liability clearing in networks is modeled as global sections of a liability sheaf on directed hypergraphs, identified as a finite-limit construction with existence and uniqueness from lattice and metric theorems on payment objects.
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Sheaf-Theoretic Transport and Obstruction for Detecting Scientific Theory Shift in AI Agents
A finite sheaf-theoretic framework ranks obstruction measures to identify when an AI agent's theory must deform within its language or extend to a new one, validated on a controlled transition benchmark.
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Heterogeneous Sheaf Neural Networks
HetSheaf applies cellular sheaves and type-conditioned restriction maps to heterogeneous graphs, plus SheafPool for basis-invariant graph-level representations, delivering competitive accuracy with substantially reduced parameter counts.
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The Cartan-Topos Protocol: A Unified Geometric and Categorical Framework for Resilient Multi-Agent Coordination
Proposes a geometric-categorical synthesis using Riemannian flows, Clifford rotors, sheaf Laplacians, and topos logic for resilient multi-agent consensus and planning.