Profinite Cosheaves Valued in Pro-regular Categories
Jiacheng Tang
TL;DR
Profinite coproducts indexed by profinite spaces are generalized to cosheaves valued in pro-regular categories, extending Wilkes's cosheaf-bundle correspondence from profinite modules to profinite groups. The work proves that CoSh(C) = Pro(CoSh(C)fin) for C = Pro(D), and that the global cosections functor (A,X) ↦ A(X) commutes with inverse limits, enabling a canonical extension of finite coproducts to the profinite setting. It shows coherence transfer when D is coherent and yields a profinite cosheaf–bundle equivalence for groups via D = Grp_fin, i.e. CoSh(X, PGrp) = Grp(Pro_{/X}). The results provide a canonical pro-regular categorical framework for profinite cosheaves with concrete equivalences between cosheaf categories and pro-categories of finite objects, strengthening the theory of profinite coproducts and their global sections.
Abstract
We prove that the category of profinite cosheaves valued in a pro-regular category (satisfying mild assumptions) is itself a pro-regular category. As a corollary, we extend Wilkes's cosheaf-bundle equivalence from profinite modules to profinite groups.