The Categorical Lift

The p-adic Langlands correspondence is a bijection: it matches smooth p-adic representations of GL₂(Qₚ) with certain modules over (φ,Γ)-algebras, one object to one object. The bijection has been known, but a bijection between objects carries no information about morphisms — it says two collections have the same size without saying how the collections are structured.

The categorification lifts the bijection to a fully faithful functor — an embedding of the entire derived category of representations into the category of Ind-coherent sheaves on (φ,Γ)-module stacks. Not just objects correspond; the morphisms between objects, the extensions, the higher homological algebra all embed faithfully. The structure of one side is a substructure of the other.

The construction is a fully faithful functor, not an equivalence. The representation-theoretic side embeds into the geometric side, but the geometric side is larger — it contains objects that do not correspond to representations. The asymmetry is meaningful: the geometric world is richer than the representation-theoretic world, and the functor identifies exactly which geometric objects have representation-theoretic meaning.

The structural observation: a correspondence between individual objects is promoted to a correspondence between entire mathematical universes, preserving all relationships between objects. The lift from bijection to functor is the difference between knowing that two libraries have the same number of books and knowing that every reference between books in one library has a corresponding reference in the other. The categorical structure — the morphisms, not just the objects — is the content of the Langlands program at the deepest level.