Profinite Cosheaves Valued in Pro-regular Categories

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.

💡 Research Summary

The paper investigates the categorical properties of profinite cosheaves whose values lie in a pro‑regular category. A pro‑regular category C is taken to be the pro‑completion Pro(D) of a small regular category D, and it is assumed that inverse limits in C commute with finite colimits—a condition automatically satisfied for such pro‑completions. The authors first develop a general theory of cosheaves over profinite spaces X (objects of the category Pro of profinite sets) with values in any category C satisfying the above limit‑colimit commutation. A precosheaf is defined as a functor from the poset of clopen subsets Oc(X) to C; a cosheaf is a precosheaf that preserves the initial object and binary disjoint unions, i.e. it sends disjoint clopen subsets to coproducts. Cosheafification (−)cosh is shown to be a right adjoint to the inclusion of cosheaves into precosheaves, and it is exact because inverse limits in C are exact. The constant cosheaf functor Δ: C → CoSh(X, C) is right adjoint to the global cosections functor A ↦ A(X).

Next, the Grothendieck construction is applied to the functor F: Pro → Cat, X ↦ CoSh(X, C), producing the category CoSh(C) whose objects are pairs (A, X) with X a profinite space and A a C‑valued cosheaf on X. Using a general result on Grothendieck constructions (Prop. 2.9) and the exactness of cosheafification, the authors prove that CoSh(C) has all limits, and they give an explicit description of inverse limits: if (A_i, X_i) is an inverse system, then X = lim← X_i in Pro and A = lim← f_i^* A_i in CoSh(X, C), where f_i: X → X_i are the projection maps. A key technical lemma (Lemma 2.11) shows that when each X_i is finite, the inverse limit of the underlying precosheaves is already a cosheaf, and the global sections satisfy A(U) = lim← A_i(f_i(U)).

The central theorem (Theorem 3.7) states that CoSh(C) is equivalent to the pro‑completion of its subcategory of “finite objects” CoSh(C)_fin, i.e. CoSh(C) ≅ Pro(CoSh(C)_fin). Here CoSh(C)_fin consists of those (A, X) with X finite and each costalk A_x belonging to D (rather than merely to C). The proof follows a recognition principle:

1. Lemma 3.4 shows every object of CoSh(C) is an inverse limit of objects from CoSh(C)_fin.
2. Lemma 3.5 proves that if the transition maps in an inverse system are epimorphisms in CoSh(C)_fin, then the induced projection maps in the limit are epimorphisms in CoSh(C).
3. Lemma 3.6 establishes that for such systems the canonical map
   lim← Hom_CoSh(C)(A_i, B) → Hom_CoSh(C)(lim← A_i, B)
   is surjective (and injective when the projections are epic).
4. Proposition 3.3 together with Lemma 3.1 shows that the ambient pro‑regular category C itself can be written as an inverse limit of objects with epic transition maps; consequently the same holds for CoSh(C)_fin.

These ingredients guarantee that CoSh(C) satisfies the defining universal property of a pro‑category, yielding the claimed equivalence.

Several corollaries are derived. If D is coherent, then CoSh(C) is also coherent (Corollary 3.8). The global cosections functor CoSh(C) → C commutes with inverse limits (Corollary 3.9), making it the unique extension of the finite‑object global sections functor. Finally, by taking D to be the category of finite groups, C becomes the category PGrp of profinite groups. The authors obtain a direct analogue of Wilkes’s cosheaf‑bundle equivalence for modules: for any profinite space X, the category of profinite‑group‑valued cosheaves CoSh(X, PGrp) is equivalent to the category of group objects in the slice Pro/X (Remark 3.10). This extends Wilkes’s result from profinite modules to profinite groups.

In summary, the paper provides a robust categorical framework for profinite cosheaves valued in pro‑regular categories, proving that the resulting category is itself pro‑regular and preserving many desirable exactness and regularity properties. The work unifies and generalizes earlier results on profinite modules, opens the door to analogous constructions for other algebraic structures (e.g., profinite rings, algebras), and suggests further applications in areas where inverse limits over profinite spaces interact with algebraic data, such as Galois cohomology, profinite homotopy theory, and the study of étale fundamental groups.