Unifying Gauge Symmetries and Spacetime Differential Structures Using ∞-Categories

This paper proposes a unified framework based on higher category theory, incor?porating gauge symmetries and spacetime differential structures into the languagesystem of ∞-categories. By elevating classical gauge groups to ∞-groups and usingsmooth ∞-stacks to describe spacetime geometry, a fiber ∞-category is constructedto unify gauge fields and manifold structures. Further categorification of the Yang?Mills action and path integrals reveals the deep connections between higher-ordersymmetries and geometric dualities. Addressing potential gaps between theoret?ical formalization and physical practice, this paper supplements explanations forthe categorical complexity of the mass gap, proposes computability schemes forpath integrals, and explores experimental verification possibilities in topologicalmaterials. Rigorous validation is conducted based on Jacob Lurie’s higher algebraframework, proposing a new direction for quantum gravity unification.