Sheaf Topos Theory: A powerful setting for Lagrangian Field Theory
Grigorios Giotopoulos
TL;DR
The paper addresses the challenge of rigorously describing classical field theory, including bosonic and fermionic fields, local Lagrangians, gauge symmetries, and topological sectors, within a unified mathematical framework that avoids problematic infinite-dimensional manifolds. It develops a topos-theoretic, functor-of-points approach by modeling field spaces as generalized spaces, namely sheaves on categories of probes such as $CartSp$ and $SupCartSp$, with Lagrangians and variational calculus realized as maps of smooth or super smooth sets and the Euler–Lagrange equations captured via a variational cotangent bundle. Extending this framework to fermions via the category of super Cartesian probes yields a robust description of fermionic field spaces, including the necessity of multiple odd coordinates for nontrivial Lagrangians, and extends jet bundle formalisms to the super smooth setting. The outlook then sketches infinitesimal thickening and higher gauge generalizations through thickened probes and smooth $\infty$-groupoids, enabling a rigorous variational calculus for flux-quantized higher gauge fields and their classifying spaces, with detailed development to come in subsequent works.
Abstract
We provide an introductory exposition to the sheaf topos theoretic description of classical field theory motivated by the rigorous description of both $\bf{(i)}$ the variational calculus of (infinite dimensional) field-theoretic spaces, and $\bf(ii)$ the non-triviality of classical fermionic field spaces. These considerations naturally lead to the definition of the sheaf topos of super smooth sets. We close by indicating natural generalizations necessary to include to the description of infinitesimal structure of field spaces and further the non-perturbative description of (higher) gauge fields.