As a philosophy of physics student in the 1990s, I approached particle physics with a desire to understand the types of thing which exist at the most fundamental physical level. Already converted to structural realism, I sought to find the mathematical structures used to represent elementary particles. What I found, in the physics textbooks, were some structures, but also a turgid and confusing array of calculational recipes; in the mathematical texts, meanwhile, I found structures such as fibre bundles and gauge connections, but often presented in frustrating abstraction from the realities of theoretical particle physics. Although Andrzej Derdzinski’s 1992 monograph, Geometry of the Standard Model of Elementary Particles, expounded the structure of the firstquantized standard model, Derdzinski’s work was difficult for a non-mathematician to grasp, and given that it promoted an unusual approach, it was often difficult to see the relationship between the structures in Derdzinski’s book, and the things which particle physicists write about. The current work has therefore grown from my own exasperated failure, in the 1990s, to understand the mathematical structure of particle physics.