Higher Structures from Simple Lattice ModelsRingrose, Noah Fusion categoriesQuantum field theoryTopological DefectsNon-invertible symmetriesTQFTCondensed matter physicsConformal Field TheoryGauge symmetryLattice modelsStatistical mechanicsMathematicsB SRecent developments have revealed that global symmetries of physical systems can be described and reconstructed from topological defects of various codimensions, which constrain the dynamics of the theory \cite{Gaiotto_2015}. These defects can be locally deformed and fused according to the discrete combinatorial data of a structure known as a fusion category—without affecting the underlying physics. This realization has sparked a surge of new results and unprecedented interdisciplinary collaboration across mathematics \cite{décoppet2024classificationfusion2categories,freed2024topological}, high-energy theory \cite{gagliano2025higherrepresentationsquarkconfinement,cordova2024particlesolitondegeneracy2dquantum}, and condensed matter physics \cite{kawagoe2024levinwengaugetheoryentanglement,seifnashri2025gaugingnoninvertiblesymmetrieslattice,Tan:2022vaz}. Structures and symmetries that were once seen as niche or exotic are now being understood as part of a broader unifying mathematical framework. While category theory and higher structures often lie outside the standard training of physicists \textemdash and many of the first examples of categorical symmetries are very unfamiliar to mathematicians \textemdash these modern methods emerge naturally in some of the simplest physical models, and the underlying mathematics is surprisingly intuitive. In this thesis, we explore how the language of topological defects and fusion categories arises directly from studying the simplest and most fundamental classical lattice model: the 2D Ising model. Beginning with the physics of the early nineteenth century, we develop from the ground up the theories of classical and quantum statistical mechanics, duality, gauge symmetry, conformal field theory, and non-invertible symmetries. My goal is to demonstrate how topological and categorical structures are not merely abstract and esoteric mathematical formalisms, but are in fact necessary to study even the most familiar and ubiquitous physical theories.Adrian Ocneanu, Thesis SupervisorLuen-Chau Li, Thesis Honors Advisoropen_access2025-04-17T05:50:47Zhttps://honors.libraries.psu.edu/catalog/9849nmr5730