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The equivalence between rational G-sheaves and rational G-Mackey functors for profinite G

For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of G-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed. This extends the classification of rational G-Mackey functors for finite G of Th\'{e}venaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second author's thesis.

The second author gratefully acknowledges support from the Engineering and Physical Sciences Research Council under Grant 1631308. The authors wish to thank the anonymous referee for their many useful suggestions that improved the paper.

Introduction
The classification of rational Mackey functors for finite groups is well-known and highly useful. An algebraic version of this result is given by Thévenaz and Webb [TW95]. A version more suited to algebraic topology can be found in work of Greenlees and May [GM95,Appendix A]. In this case the classification is given as an equivalence of categories where the product runs over G-conjugacy classes of subgroups of G. Greenlees and May proved this result in order to classify rational G-spectra (for finite G) in terms of an algebraic model. A gentle introduction to classifying rational G-spectra (and similar results) can be found in work of the first author and Kędziorek [BK20].
Profinite groups (compact Hausdorff totally-disconnected topological groups) are an important class of topological groups that contain all finite groups. Examples of where profinite groups occur include the automorphism groups of Galois extensions of fields, the étale fundamental groups of algebraic geometry and the Morava stabilizer group S n from chromatic homotopy theory. The standard example is the p-adic integers Z ∧ p for p a prime. The main result of this paper is an extension of the classification of rational Mackey functors to profinite groups. Here SG denotes the space of closed subgroups of G, see Definition 2.8 for the topology. A Weyl-G-sheaf over SG is a G-equivariant sheaf over SG such that the stalk at a closed subgroup K G is K-fixed. Hence, the stalk at K has an action of the Weyl group W G K. This result appears as Theorem 4.8 in the main body.
The reason why Mackey functors for finite groups have a simpler description is that SG is a discrete space when G is finite. In which case, a Weyl-G-sheaf over SG is a product of Q-modules with actions of the Weyl groups. Choosing representatives for conjugacy classes gives the description (1) above. Furthermore, the rational Burnside ring for a profinite group is isomorphic to the ring of continuous functions from SG/G to Q. Hence, when G is finite, the rational Burnside ring is simply a product of copies of Q, indexed over the set of conjugacy classes of subgroups of G.
Our next result, see Lemma 2.17, extends a vital identity on rational G-Mackey functors from the finite case to the profinite case. In the finite setting, the restriction map induces an isomorphism where e H K is the idempotent in the rational Burnside ring of H corresponding to the conjugacy class of K H. To extend this to the profinite case, we use the idempotents e H N,K and e K N,K of Definition 2.13, which are analogous to e H K and e K K . The following is Proposition 2.21 and relates the equivariance of the stalks to that of representing sections. This proposition and the preceding lemma are key to proving the equivalence of categories of Theorem A.
Proposition C. Let E be a Weyl-G-sheaf over SG and K a closed subgroup of G. A germ over K can be represented by an N K-equivariant section over a neighbourhood of K, for some open normal subgroup N of G.
This work is part of the second author's PhD thesis, [Sug19], completed under the supervision of the first author. That thesis applies the main result of this paper to obtain a classification of rational G-spectra for profinite G in terms of chain complexes of Weyl-G-sheaves over SG, see [Sug19, Chapters 3 and 6]. This extends the work of the first author on the case where G is the p-adic integers, [Bar11]. A further application of the classification of rational G-Mackey functors in terms of sheaves is a calculation of the injective dimension of these categories in terms of the Cantor-Bendixson rank of the space SG, see [Sug19, Chapter 8].
1.1. Organisation. We define the categories involved in Section 2 and construct the two functors of the classification in Section 3. We prove directly that the functors give an equivalence in Section 4. We end the paper with some examples and the relation between G-sheaves over SG and Weyl-G-sheaves over SG, see Section 5.

Mackey functors and sheaves
We introduce the definitions and categories that are used in the paper.
2.1. Basic facts on profinite groups. We give a few reminders of useful facts on profinite groups. More details can be found in Wilson [Wil98] or Ribes and Zalesskii [RZ00].
A profinite group is a compact, Hausdorff, totally disconnected topological group. A profinite group G is homeomorphic to the inverse limit of its finite quotients: The limit has the canonical topology which can either be described as the subspace topology on the product or as the topology generated by the pre-images of the open sets in G/N under the projection map G → G/N , as N runs over the open normal subgroups of G. Any open subgroup H contains an open normal subgroup, the core of H in G, defined as the following intersection. Note that as H has finite index in G, this intersection has only finitely many distinct terms. Core We can also define a profinite topological space to be a Hausdorff, compact and totally disconnected topological space. As with profinite groups, such a space is homeomorphic to the inverse limit of a filtered diagram of finite spaces. Moreover, a profinite topological space has a closed-open basis.
2.2. Mackey functors for profinite groups. There are many equivalent definitions of Mackey functors for a finite group G. Our preferred version is a collection of abelian groups indexed over the subgroups of G with induction (transfer), restriction and conjugation maps satisfying a list of axioms. In turn, there are several (inequivalent) extensions to profinite groups. Since we have in mind applications to algebraic topology and the work of Fausk [Fau08], we choose a generalisation that emphasises the role of the open subgroups.
The following definition can be found in Bley    (2) For L K H open subgroups of G and g, h ∈ G, there are composition rules The first two are transitivity of induction and restriction. The last is associativity of conjugation.
This condition is known as the Mackey axiom. for each open subgroup H G, which commute with the induction, restriction and conjugation maps. We use Mackey Q (G) to denote this category. See Thévenaz and Webb [TW95] for an introduction into Mackey functors for finite groups. The case of rational Mackey functors is also discussed in work of the first author with Kędziorek [BK20]. Alternative definitions of Mackey functors in a general setting are discussed in Lindner [Lin76, Definition 1].
We have some standard examples similar to those for finite groups.  In the case of finite groups, another common example is the fixed point Mackey functor. A profinite analogue exists, but one needs to be more careful over the action of G on a Q-module.   H-sets (that is, finite discrete topological spaces with a continuous action of H). The multiplication is given by the product of H-sets.
Restriction and induction between A Q (H) and A Q (K) are given by the usual restriction and induction of sets with group actions. Moreover, the restriction maps are maps of rings. Conjugation is induced from precomposition with the conjugation homomorphism of groups.
We also have the relation, known as Frobenius reciprocity, between induction and the action of 2.3. Burnside ring idempotents. As with finite groups, there is a description of the rational Burnside ring in terms of continuous maps out of the space of subgroups of G. We will use this structure repeatedly, so we examine the result in detail and use it to describe idempotents of the Burnside ring.
Definition 2.8. For G a profinite group, the space of closed subgroups of G, SG, is the topological space where S(G/N ) is a finite space with the discrete topology. For N N ′ , the corresponding map of the limit diagram is The limit is topologised as either the subspace of the product or as the coarsest topology so that the projection maps are continuous. The group G acts continuously on SG by conjugation, g ∈ G sends a closed subgroup K to gKg −1 = g K.
Following Gartside and Smith [GS10, Lemma 2.14], we can give a compact-open basis for this topology.
These sets are compatible with the G-action: Since any open subgroup of G has form N K for N some open normal subgroup of G and K a closed subgroup of G, we can rewrite the basis to consist of the sets If we fix K and allow N to vary, these sets give a neighbourhood basis for K ∈ SG. We further define It is closed as O G (N, K) is compact. We will also make use of the equality Theorem 2.10. For G a profinite group, there is an isomorphism of commutative rings Proof. This is Dress [Dre71, Theorem B.2.3(a)]. As with finite groups, the isomorphism maps [G/H] to the function which sends a conjugacy class of closed subgroup K to |(G/H) K |.
Proof. Since any G/N -set is a G-set by precomposing with the quotient G → G/N , we have the given map of rings, which is injective. Surjectivity follows as a continuous action of G on a finite set factors through some finite quotient of G.
Remark 2.12. Similar to Pontryagin duality, the previous lemma and theorem imply that the canonical map induced by the surjections SG/G → S(G/N )/(G/N ) is an isomorphism. This type of isomorphism holds for any profinite space X. The difficult step is proving surjectivity. This comes from the fact that any map X → Q must have finite image and hence factor through some finite quotient of X. We relate our closed-open basis for SG to idempotents of C(SG/G, Q) and the additive basis of that ring. For the formula we will use the Moebius function µ for pairs of subgroups D K where c i is the number of strictly increasing chains from D to K of length i. Note that a chain of the form D < K has length 1 and, by convention, µ(K, K) = 1.
Proposition 2.14. For G a profinite group, with open normal subgroup N and closed subgroup K Proof. The sum is finite as we range over open subgroups of G that contain N . The method is to reduce the calculation to the case of finite groups, and use the formula for idempotents of the rational Burnside ring of the finite group G/N from Gluck [Glu81, Section 2]. The reduction comes from the following commuting diagrams, where p N : SG → S(G/N ) sends H to HN/N . The second diagram uses the homeomorphism of G-spaces: If we replace p N by p N : SG/G → S(G/N )/(G/N ) the two diagrams above remain commutative. The first diagram and the formula for idempotents gives The second diagram completes the proof.
The coefficients are constant under certain conjugations. More precisely, for K J G open subgroups of G and a ∈ N G (J), if we define Proposition 2.15. Suppose X is a profinite space and M is a C(X, Q)-module, then M determines a sheaf of Q-modules over X.
Proof. Let B be an open-closed basis for X. Given a module M , we define a sheaf F by giving its values on B. Let U ∈ B and define the characteristic function of U to be the map which sends elements of U to 1 and the rest of X to 0. This map, which is denoted e U ∈ C(X, Q), is continuous as U is both open and closed. We define F (U ) = e U M . For V ⊆ U in B, we define the restriction map to be the projection (where V c is the complement of V in X) Paraphrasing the corollary, we see that a G-Mackey functor is a collection of sheaves, one for each open subgroup of G. The rough idea of the equivalence is to patch parts of these sheaves together to construct a G-equivariant sheaf. It will take some time to make this rigorous, but we can give a useful relation between the various sheaves M (H). This relation is an extension of a result for finite groups, as referenced in the proof, see also Equation (2) of the introduction.

Equivariant sheaves.
We begin with the definition of a G-equivariant sheaf over a profinite G-space X where G is a profinite group. The second author gives two equivalent definitions in [Sug19], see also [BS20]. We work with just one for brevity.
Definition 2.18. A G-equivariant sheaf of Q-modules over X is a map of topological spaces p : E → X such that: (1) p is a G-equivariant map p : E → X of spaces with continuous G-actions, (2) (E, p) is a sheaf space (étale space) of Q-modules, (3) each map g : p −1 (x) → p −1 (gx) is a map of Q-modules for every x ∈ X, g ∈ G.
We will write this as either the pair (E, p) or simply as E. We call E the total space, X the base space and p the structure map.
We call p −1 (x) the stalk of E at x and denote it E x . An element of a stalk is called a germ.
Note that points (1) and (2) give a map of sets for point (3), but they do not imply that it is a map of Q-modules. Given a G-equivariant sheaf (E, p) over X and x ∈ X, the stalk E x (equipped with the discrete topology) has a continuous action of stab G (x).
As in the non-equivariant case, given an open subset U ⊆ X the space of (continuous) sections has an addition operation (defined stalk-wise). The sections are not required to be G-equivariant. Allowing U to vary defines a functor from the set of open subsets of X to Q-modules. Moreover, one can show that colim as in the non-equivariant setting.
Given a section s : U → E and g ∈ G, we can define We also need a result which can be described as saying that a section of a G-equivariant sheaf over a profinite G-space is "locally sub-equivariant". See [Sug19, Section 4.3] for related results. for some N M ∩ M ′′ . The section s |W is N K-equivariant by our earlier argument as gw ∈ W and gs(w) ∈ V ⊆ s(U ) for all g ∈ N K and w ∈ W .

The functors
In this chapter we will construct a correspondence between rational G-Mackey functors and Weyl-G-sheaves over SG. We shall explicitly construct functors between the categories, see Theorems 3.5 and 3.13. In Theorem 4.8 we will see that these functors are equivalences of categories.
3.1. Weyl-G-sheaves determine Mackey Functors. We define a functor: We will not need the input sheaf to be a Weyl-G-sheaf, a detail we return to in Subsection 5.2 Let T ⊂ H be a left transversal of K in H. For β a section of SK, let β be the extension by zero to SH. We define the induction map by We prove that the induction functor is well defined in Lemma 3.2. The structure maps compose and interact appropriately by Lemma 3.3. Lemma 3.4 proves that the Mackey axiom holds. Proof. Let hK = h ′ K be two representatives for the same coset. Since the section β is K-fixed, h * β = h ′ * β as h and h ′ differ by an element of K. It follows that I H K is independent of the choice of T . The sum of h * β over h ∈ T is H-equivariant as T is a transversal. Proof. The part of most interest is that the induction maps are transitive and equivariant. The first follows from the fact that given open subgroups J K H, one can combine a transversal of J in K with a transversal of K in H to get a transversal of J in H. This gives the transitivity. For equivariance, the result follows as one can conjugate a transversal to get a transversal of the conjugate.
We note that SH is invariant under the action of N G H G, so E(SH) has a continuous action of N G H.
It remains to show the Mackey axiom, which is a direct calculation.
where the extension by zero in the third term is with respect to H and in the in the fourth term it is with respect to J. If we start from the other direction we have: proving that the two sides coincide.
Similar arguments to the above show that a map of sheaves E → E ′ induces a map of G-Mackey functors Mackey(E) → Mackey(E ′ ). We summarise this section in the following theorem.
Theorem 3.5. Let G be a profinite group. If E is a G-sheaf of Q-modules over SG, then Mackey(E) from Construction 3.1 is a Mackey functor and the assignment is functorial. Hence, there is a functor

Mackey Functors determine Weyl-G-sheaves.
We construct a functor in the opposite direction, from rational G-Mackey functors to Weyl-G-sheaves of Q-modules over SG. We start by constructing the stalks of the Weyl-G-sheaf. We begin our construction of a rational G-equivariant sheaf (E, p) over SG from a Mackey functor M . We first define E and p.
Since C h = Id M(H) , we have the desired equivariance. We also see that m K is fixed by N H (K).
We can restrict a section s = θ N K (m) coming from m ∈ e N K N,N K M (N K) to a section over (N, N K). (N, N K)) for varying open and normal N , closed K, and s = θ N K (m) for m in e N K N,N K M (N K), define a topology on Sheaf(M ) by Lemma 3.8.

The sets s(O G
The G-action is continuous by Lemma 3.9 and the projection map p is a G-equivariant local homeomorphism by Proposition 3.10. Lemma 3.11 shows that (Sheaf(M ), p) is a sheaf of Q-modules. Lemma 3.12 completes the construction by showing that we have a Weyl-G-sheaf. Now we show that the intersection of two basis sets is a union of basis elements. Take sections t 1 = θ N1K1 (m 1 ) and t 2 = θ N2K2 (m 2 ) and let We construct a t(O G (N, N K)) satisfying Given such an x, we let L = p(x), which is a closed subgroup of G. We see that  (N, N L). We define t to be this common refinement and see  (N, N K)) under the group action map. Then We can therefore find an open normal subgroup N and representatives such that These sections have N -invariant domains and are obtained from an N -fixed module M (N L). We can conclude that the open set: is contained in pre-image of s(O G (N, N K)) under the group action map. Proposition 3.10. The projection map in Construction 3.7 is a continuous G-map and a local homeomorphism.
Proof. We first prove that p is continuous. Given a basic open subset of SG of the form O G (N, N K), we have that Take s ∈ Sheaf(M ) L and a section with t(L) = s and N ′ N . N, N K)) and is open. We now show that p is a local homeomorphism. A point s in Sheaf(M ) L has a neighbourhood of the form t(O G (N, N K)) as seen previously. Let f = p |t(OG(N,N K)) , we claim that

is a homeomorphism. Since f is bijective and continuous, we need only show that it is open.
A basic open set of t(O G (N, N K)) is a set of the form Proof. Given two sections s and t of p we can restrict ourselves to neighbourhoods of SG and E such that p is a local homeomorphism. From the construction of the topology on E it follows that on this region, s and t come from elements of some M (N K), where N is an open normal subgroup and K is a closed subgroup of G. The addition of M (N K) defines a section s + t. Similarly, the Q-action on M (N K) defines q · t for q ∈ Q.
Lemma 3.12. The G-sheaf of Q-modules (Sheaf(M ), p) defined in Construction 3.7 is a Weyl-Gsheaf. That is, g ∈ G gives maps of Q-modules We summarise this work in the following theorem. The additional statement here is that the construction is functorial. This follows from the fact that maps of Mackey functors commute with actions of Burnside rings.
Theorem 3.13. For G a profinite group, there is a functor which sends a Mackey functor M over G to a Weyl-G-sheaf denoted Sheaf(M ), as defined in Construction 3.7.
We end this subsection by noting that the maps θ H from Construction 3.7 are injective.

The equivalence
In this section we prove that the two functors Mackey and Sheaf are inverse equivalences, see Theorem 4.8.
4.1. The equivalence on sheaves. We prove that for a Weyl-G-sheaf of Q-modules F , we have an isomorphism of equivariant sheaves of Q-modules The first part is to show that we have an isomorphism on each stalk. For that, we need to know how the action of the Burnside ring on Mackey functors translates to sheaves. As s ∈ M (H) is H-fixed, the section t = C h s |SK satisfies t(hJh −1 ) = s(hJh −1 ) for J ∈ SK. It is zero outside of S(hKh −1 ). Hence, for L ∈ SH, t(L) is non-zero exactly when L ∈ S(hKh −1 ). This is equivalent to the condition hN A ∈ (H/K) L . We see that We can now see that a idempotent of A Q (H) acts by restricting a section and then extending the result by zero.  (N, N K)).
It is an isomorphism as any germ can be represented by a section that is locally sub-equivariant by Proposition 2.21 The proof requires F to be a Weyl-G-sheaf as we need the local sub-equivariance property. We now check that the preceding isomorphism is compatible with the group actions.
Lemma 4.4. Let F be a Weyl-G-sheaf of Q-modules. If K ∈ SG and g ∈ G, then the following square commutes where ψ K is the map defined in the proof of Proposition 4.3.
Proof. A germ s K ∈ Sheaf • Mackey(F ) K can be represented by an N K-equivariant section This section is also a representative for ψ K (s K ), from which the commutativity of the square follows.
Theorem 4.5. If F is any Weyl-G-sheaf of Q-modules, then the maps ψ K induce an isomorphism in the category of Weyl-G-Sheaves.
Proof. By Proposition 4.3 and Lemma 4.4, we see that they are isomorphic as G-sets. To prove that they are topologically equivalent, we need to show that they have the same sections. As both objects are sheaves, it suffices to do so locally, which amounts to considering sections that represent stalks. By Proposition 4.3, a germ over K ∈ SG of Sheaf • Mackey(F ) is represented by some element of F (O N K (N, N K)) N K . By the local sub-equivariance property of Proposition 2.21 the same is true for a stalk of F .  Since the topology was defined using the images of θ H , it follows that θ H (m) is a continuous section. Thus θ H defines a map as in the statement. Moreover, it is additive and surjective, injectivity follows from Lemma 3.14.
In the final equality we replace the extension by zero by an explicit condition on the subgroups.  which is sent to C x (m) L in the L-stalk.
The collection of x ∈ [L\H/K] such that L xKx −1 , is exactly the set hK ∈ H/K such that h −1 Lh K (the canonical isomorphism sends LxK to xK). Hence, We combine Theorems 4.5 and 4.7 and note that our constructions are compatible with maps in the two categories to obtain the main result.
Theorem 4.8. If G is a profinite group then the category of rational G-Mackey functors is equivalent to the category of Weyl-G-sheaves over SG. Furthermore, this is an exact equivalence.

5.1.
Examples. Just as non-equivariantly, one can make constant sheaves and skyscraper sheaves. We leave the details to the second author's thesis [Sug19] and the forthcoming paper by the authors [BS20]. The stalks of the constant sheaf at a Q-module B are all isomorphic to B. The sections are as for the non-equivariant constant sheaf. For an equivariant skyscraper sheaf, one picks a closed subgroup K of G and a discrete W G Kmodule C. The stalk at K is C, for g ∈ G, the stalk at gKg −1 is gC (with the expected gKg −1action), all other stalks are zero. The sections of this sheaf over U are given by a direct sum of copies of gB, with one copy for each gKg −1 ∈ U , where the sums runs over G/K. It can be illuminating to perform the reverse calculation. The fixed point Mackey functor M C is cohomological, that is, restriction followed by induction is multiplication by the subgroup index which is zero for non-trivial K and the identity map for K the trivial group. Hence, the stalks at non-trivial subgroups are zero. We then use the fact that C is discrete to complete the calculation

5.2.
Weyl Sheaves from equivariant sheaves. As mentioned, Construction 3.1 does not require the input G-sheaf to be a Weyl-G-sheaf. Hence, given a sheaf F , Sheaf • Mackey(F ) is a Weyl-Gsheaf. We can construct this operation explicitly.