Formulate a theory of fibered categories with base-induced symmetry actions

Develop a general mathematical framework for fibered categories equipped with symmetry actions arising from their base categories, specifically for the fibered category whose objects are schemes over arithmeticoids of a number field (the objects (X,L_ε)^{sch} in the category 𝔅𝓸𝓁𝒹 𝒮𝒸𝒽_L) with projection to the space of arithmeticoids (as in Equation (eq:proj-to-arithmeticoid)), so that global symmetries acting on the base (Galois, Frobenius, and multiplicative actions) are functorially realized on the total category.

Background

The paper constructs, for a fixed number field L, a family of categories of schemes over arithmeticoids arith{L}_ε and organizes them into a fibered category 𝔅𝓸𝓁𝒹 𝒮𝒸𝒽_L over the base space of arithmeticoids. Natural global symmetries—namely the Galois action, a global Frobenius, and an L^* action—act on the base and are intended to act on the total category via the projection.

While this setup is central to the arithmetic Teichmüller framework developed in the paper, the author notes that a general abstract formulation of fibered categories endowed with such base-driven symmetry actions is lacking. Establishing a rigorous general theory would provide a formal foundation for the symmetry-induced dynamics on categories of schemes over varying arithmeticoids.