Prekosmic Grothendieck/Galois Categories

Abstract

We develop a theory on duality between groups and the categories of their representations, by providing an abstract characterization of such categories. The theory has two parallel formulations, each presenting a duality analogous to one of the two that were originally proposed by Grothendieck: the one between profinite groups and the abstract categories of their continuous representations on finite sets, and the other between affine group schemes over a field and the abstract categories of their finite-dimensional linear representations. In the present theory, groups and their representations are internal to a symmetric monoidal base category, with no finiteness condition imposed. The role of a fiber functor is played by a fiber point, an abstraction of the forgetful functor with its chosen adjoint. Over each base, a group determines the category of its representations equipped with the canonical fiber point, and a pointed abstract category determines the group that represents the presheaf of automorphisms of its chosen fiber point. These constructions yield mutually quasi-inverse equivalences of $(2,1)$-categories. By introducing change of base, we gather the dualities over individual bases into a single duality in each formulation. For a pointed abstract category, fiber points of the category are in perfect correspondence with torsors under the group associated to the chosen point. As an illustration of the theory, we reformulate the Galois theory of covering spaces for nonempty, path-connected, locally path-connected, semilocally simply connected spaces.