ORCID

https://orcid.org/0009-0002-5829-3582

Date of Award

Spring 2026

Language

English

Embargo Period

5-1-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Program

Mathematics

First Advisor

Justin Curry

Committee Members

Justin Curry, Michael Robinson, Boris Goldfarb, Felix Ye, Cliff Joslyn

Keywords

Category theory, Hypergraphs, Higher Order Networks, Homology, Sheaves

Subject Categories

Geometry and Topology

Abstract

Hypergraphs are a prominent tool for representing networks with connections among three or more entities. There is an inherent flexibility that allows hypergraphs to more naturally represent certain types of networks than graphs or simplicial complexes can on their own. This flexibility, however, comes at a cost, as there is a zoo of various categories and homology theories that are applicable to hypergraphs. The first chapter of this dissertation explores various categorical perspectives on hypergraphs, focusing on what the natural notion of morphism between hypergraphs should be. It also contains an exploration of the functoriality of vertex-edge duality in these categories, as well as a novel investigation of hypergraphs as a relaxation of topological spaces. The second chapter of this dissertation focuses on the relationship between two main notions for the homology of a hypergraph: the downward closure homology and the restricted barycentric subdivision homology. Tools from finite topology are used to prove a new theorem: the homologies are equivalent if the hypergraph is closed under intersection. We leverage this result to study the hyperblock as a generalized persistence module and provide a novel topological invariant for hypergraphs that is more discerning than either homology theory on its own. The third chapter of this dissertation re-frames the theory of sheaves over posets, introducing the language of upset sheaves for the first time, to provide new results and applications of sheaves on hypergraphs. Specifically, we show that the map relating various poset completions of a hypergraph is homotopy initial, in a categorical sense, and then develop a new theory of ontological type-checking using sheaves valued in the category of posets. Ontologically valid typings of events are shown to be equivalent to lax sections of these ontology sheaves. A generalization of Robinson's consistency radius is given in this lax setting to quantify how far a given sentence is from being ontologically coherent. The Grothendieck construction is applied to the ontology sheaf to provide a new perspective on how to compute the consistency radius and various lines of future research are outlined.

License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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