Theory and Applications of Categories, Vol. 22, No. 16, 2009, pp. 388–400.

EPIMORPHIC COVERS MAKE R_{G}^{+} A SITE, FOR PROFINITE G

DANIEL G. DAVIS

Abstract. Let G be a non-finite profinite group and let G−Sets_{df} be the canonical site of finite discrete G-sets. Then the category R_{G}^{+}, defined by Devinatz and Hopkins, is the category obtained by considering G−Sets_{df} together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R_{G}^{+} is a site when equipped with the pretopology of epimorphic covers. We point out that presheaves of spectra on R_{G}^{+} are an efficient way of organizing the data that is obtained by taking the homotopy fixed points of a continuous G-spectrum with respect to the open subgroups of G. Additionally, utilizing the result that R_{G}^{+} is a site, we describe various model category structures on the category of presheaves of spectra on R_{G}^{+} and make some observations about them.

1. Introduction

Let G be a profinite group that is not a finite group. Let R_{G}^{+} be the category with objects all finite discrete left G-sets together with the left G-space G. The morphisms of R_{G}^{+} are the continuous G-equivariant maps. Since G is not finite, the object G in R_{G}^{+} is very different in character from all the other objects of R_{G}^{+}. In this paper, we show that R_{G}^{+} is a site when equipped with the pretopology of epimorphic covers.

As far as the author knows, the category R_{G}^{+} is first defined and used in the paper [DH04]. Let G_{n} be the profinite group S_{n} ⋊ Gal(𝔽_{p^{n}}/𝔽_{p}), where S_{n} is the nth Morava stabilizer group. In [DH04, Theorem 1], the authors construct a contravariant functor - that is, a presheaf -

F: (R_{G_{n}}^{+})^{op} → (E_{∞})_{K(n)},

to the category (E_{∞})_{K(n)} of K(n)-local commutative S-algebras (see [EKMM97]), where K(n) is the nth Morava K-theory (see [Rud98, Chapter 9] for an exposition of K(n)). The functor F has the properties that, if U is an open subgroup of G_{n}, then F(G_{n}/U) = E_{n}^{dhU}, and F(G_{n}) = E_{n}, where E_{n} is the nth Lubin-Tate spectrum (for salient facts about E_{n} and its importance in homotopy theory, see [DH95, Introduction]), and E_{n}^{dhU} is a spectrum that behaves like the U-homotopy fixed point spectrum of E_{n} with respect to the continuous U-action. Since Hom_{R_{Gn}^{+}}(G_{n}, G_{n}) ≅ G_{n}, functoriality implies that G_{n} acts on E_{n} by maps of commutative S-algebras. In Section 5, we will give several related examples of presheaves of spectra that illustrate the utility of the category R_{G}^{+}.

Received by the editors 2006-10-31 and, in revised form, 2009-07-23.

Transmitted by Brooke Shipley. Published on 2009-07-27. Reference [Dav06a] updated 2009-10-05.

2000 Mathematics Subject Classification: 55P42, 55U35, 18B25.

Key words and phrases: site, profinite group, finite discrete G-sets, presheaves of spectra, Lubin-Tate spectrum, continuous G-spectrum.

\begin{matrix}c \\ ◯ \end{matrix} Daniel G. Davis, 2009. Permission to copy for private use granted.

EPIMORPHIC COVERS MAKE R_{G}^{+} A SITE, FOR PROFINITE G  389

The pretopology of epimorphic covers on a small category C is the pretopology K given by all covering families {f_{i}: C_{i} → C| i ∈ I} such that ϕ: ∐_{i∈I} C_{i} → C is onto, where C_{i}, C ∈ C, f_{i} ∈ Mor_{C}(C_{i}, C), and I is some indexing set. (Of course, one must prove that these covering families actually give a pretopology on C.) We note that we do not require that ϕ be a morphism in C; for our purposes, C = R_{G}^{+} and we only require that ϕ be an epimorphism in the category of all G-sets (so that ϕ does not have to be continuous). This assumption is important for our work, since, for example, G ∐ G is not in R_{G}^{+}. Also, recall that the pretopology K is a familiar one. For example, for a profinite group G, K is the standard basis used for the site G − Sets_{df} of finite discrete G-sets ([Jar97, pg. 206]).

Note that R_{G}^{+} is built out of the two subcategories G−Sets_{df} and the groupoid G. Since each of these categories is a site via K (for G, this is verified in Lemma 2.2 below), it is natural to wonder if R_{G}^{+} is also a site via K. Our main result (Theorem 3.1) verifies that this is indeed the case.

As discussed earlier, F is a presheaf of spectra on the site R_{G_{n}}^{+}. More generally, there is the category PreSpt(R_{G}^{+}) of presheaves of spectra on R_{G}^{+}. In [Dav06b], the author showed that, given a continuous G-spectrum Z, then, for any open subgroup U of G, there is a homotopy fixed point spectrum Z^{hU}, defined with respect to the continuous action of U on Z. In Examples 5.5 and 5.6, we see that there is a presheaf of spectra on R_{G}^{+} that organizes in a functorial way the following data: Z, Z^{hU} for all U open in G, and the maps between these spectra that are induced by continuous G-equivariant maps between the G-spaces G and G/U. Thus, PreSpt(R_{G}^{+}) is a natural category within which to work with continuous G-spectra.

The homotopy fixed points referred to above use a certain model category structure on the category PreSpt(G−Sets_{df}) of presheaves of spectra on G−Sets_{df} (see Section 5 and [Dav06b, Section 3] for details). Thus, in Section 5, we describe various model category structures on PreSpt(R_{G}^{+}), and we make some observations about these model structures and the restriction functor i_{∗}: PreSpt(R_{G}^{+}) → PreSpt(G − Sets_{df}) (see Definition 5.8), where the target category has the aforementioned model structure. It is our hope that one or some of these model structures on PreSpt(R_{G}^{+}) can be useful for the theory of homotopy fixed points for profinite groups, though we have not yet found any such applications.

The main result of this paper naturally motivates further work in exploring the site R_{G}^{+}. We give a brief discussion of some of the things that one might consider. The Grothendieck topos of sheaves on G − Sets_{df} is equivalent to the category of discrete G-sets, a category of continuous G-spaces. For some time, the author worked to show that the Grothendieck topos on the site R_{G}^{+} was similarly equivalent to a category of topological G-spaces. (To be more precise, the author hoped to obtain some category related to pro-discrete G-spaces.) However, the author concluded that there was no such equivalence.

As noted above in several ways, the site R_{G}^{+} is closely related to the site G−Sets_{df}. In particular, each site has a topology given by the pretopology K of epimorphic covers. However, there is an important difference between R_{G}^{+} and G−Sets_{df}: the latter category is closed under pullbacks, but it is easy to see that R_{G}^{+} does not have all pullbacks (see the discussion just after Theorem 3.1).

In a category with pullbacks, the canonical topology, the finest topology in which every representable presheaf is a sheaf, is given by all covering families of universal effective epimorphisms (see Expose IV, 4.3 of [Dem70]). This implies that G−Sets_{df} is a site with the canonical topology, when equipped with pretopology K. However, due to the lack of sufficient pullbacks, we cannot conclude that K gives R_{G}^{+} the canonical topology. Thus, it is natural to ask if the topology on R_{G}^{+} is canonical. This is not an easy question to answer: a preliminary step is to determine whether or not the topology is subcanonical, and the author is currently working to solve this problem.

In Section 4, we exhibit a second pretopology on R_{G}^{+} that generates a topology that is strictly coarser than that given by K. We do not know if this second pretopology has any advantages over K.

Acknowledgements. When I first tried to make R_{G}^{+} a site, and was focusing on an abstract way of doing this, Todd Trimble helped me get started by suggesting that I extend K to all of R_{G}^{+} and by pointing out Lemma 2.1. I thank Paul Goerss for discussions about this material. Also, I appreciate various conversations with Christian Haesemeyer about this work (and some of the work mentioned above that is motivated by the main result). I thank the referee for a variety of helpful comments. Part of this paper was written during a very pleasant year spent visiting the math department of Wesleyan University; I thank the department for its hospitality.

2. Preliminaries

Before we prove our main result, we first collect some basic facts which will be helpful later. As stated in the Introduction, G always refers to an infinite profinite group. (If the profinite group G is finite, then R_{G}^{+} = G−Sets_{df} and there is nothing to prove.)

2.1. Lemma. Let f: C → G be any morphism in R_{G}^{+} with C ≠ ∅. Then C = G.

Proof. Choose any c ∈ C and let f(c) = γ. Choose any δ ∈ G. Then the G-equivariance of f implies that δ = (δγ^{−1})γ = (δγ^{−1}) · f(c) = f((δγ^{−1}) · c). Thus, f is onto and |im(f)| = ∞, so that C cannot be a finite set.

2.2. Lemma. For a topological group G, let G be the groupoid with the single object G and morphisms the G-equivariant maps G → G given by right multiplication by some element of G. Then G is a site with the pretopology K of epimorphic covers.

Proof. Any diagram G →^{f} G ←^{g} G, where f and g are given by multiplication by γ and δ, respectively, can be completed to a commutative square

\xymatrix{G \ar[r]^{f′} \ar[d]^{g′} & G \ar[d]^{g} \\ G \ar[r]^{f} & G,}

where f′ and g′ are given by multiplication by δ^{−1} and γ^{−1}, respectively. This property suffices to show that G is a site with the atomic topology, in which every sieve is a covering sieve if and only if it is nonempty. It is easy to see that the only nonempty sieve of G is Mor_{G}(G, G) itself. Thus, the only covering sieve of G is the maximal sieve. Since every morphism of G is a homeomorphism, in the pretopology K, the collection of covers is exactly the collection of all nonempty subsets of Mor_{G}(G, G). Then it is easy to check that K is the maximal basis that generates the atomic topology.

Observe that if f: G → G is a morphism in R_{G}^{+}, then by G-equivariance, f is the map given by multiplication by f(1) on the right. As mentioned earlier, we have

2.3. Lemma. The category G−Sets_{df}, a full subcategory of R_{G}^{+}, is closed under pullbacks.

Proof. The pullback of a diagram in G−Sets_{df} is formed simply by regarding the diagram as being in the category T_{G} of discrete G-sets. The category T_{G} is closed under pullbacks, as explained in [MM94, pg. 31].

We recall the following useful result and its proof.

2.4. Lemma. Let X be any finite set in R_{G}^{+}. We write X = ∐_{i=1}^{n} \overline{x_{i}}, the disjoint union of all the distinct orbits \overline{x_{i}}, with each x_{i} a representative. Then X is homeomorphic to ∐_{i=1}^{n} G/U_{i}, where U_{i} = G_{x_{i}} is the stabilizer in G of x_{i}.

Proof. Let f: G/U_{i} → \overline{x_{i}} be given by f(γU_{i}) = γ · x_{i}. Since X is a discrete G-set, the stabilizer U_{i} is an open subgroup of G with finite index, so that G/U_{i} is a finite set. Then f is open and continuous since it is a map between discrete spaces. Also, it is clear that f is onto. Now suppose γU_{i} = δU_{i}. Then γ^{−1}δ ∈ U_{i}, so that (γ^{−1}δ)·x_{i} = (γ^{−1})·(δ ·x_{i}) = x_{i}. Thus, γ · x_{i} = δ · x_{i} and f is well-defined. Assume that γ · x_{i} = δ · x_{i}. Then γ^{−1}δ ∈ G_{x_{i}} so that f is a monomorphism.

2.5. Lemma. Let X be a finite discrete G-set in R_{G}^{+} and let ψ: G → X be any Gequivariant function. Then ψ is a morphism in R_{G}^{+}.

Proof. As in Lemma 2.4, we identify X with ∐_{i=1}^{n} G/U_{i}. Since ψ is G-equivariant and ψ(γ) = γ · ψ(1), ψ is determined by ψ(1). Let ψ(1) = δU_{j} for some δ ∈ G and some j. Then for any γ in G, γU_{j} = (γδ^{−1}δ)U_{j} = (γδ^{−1}) · ψ(1) = ψ(γδ^{−1}), so that im(ψ) = G/U_{j}. Since X is discrete, ψ is continuous, if, for any x ∈ X, ψ^{−1}(x) is open in G. It suffices, by the identification, to let x = γU_{j}, for any γ ∈ G. Then

ψ^{−1}(γU_{j}) = {ζ ∈ G| ψ(ζ) = γU_{j}} = {ζ ∈ G| ζ · (δU_{j}) = γU_{j}}
= {ζ ∈ G| δ^{−1}ζ^{−1}γ ∈ U_{j}} = γU_{j}δ^{−1}.

Since U_{j} is open and multiplication on the left or the right is always a homeomorphism in a topological group, we see that ψ^{−1}(x) is an open set in G.

3. The proof of the main result

With the lemmas of the previous section in hand, we are ready for

3.1. Theorem. For any profinite group G, the category R_{G}^{+}, equipped with the pretopology K of epimorphic covers, is a small site.

Before proving the theorem, we make some remarks about pullbacks in R_{G}^{+} and how this affects our proof. In a category C with sufficient pullbacks, to prove that a pretopology is given by a function K, which assigns to each object C a collection K(C) of families of morphisms with codomain C, one must prove the stability axiom, which says the following: if {f_{i}: C_{i} → C| i ∈ I} ∈ K(C), then for any morphism g: D → C, the family of pullbacks

{π_{L}: D ×_{C} C_{i} → D| i ∈ I} ∈ K(D).

Let us examine what this axiom would require of R_{G}^{+}.

3.2. Example. The map G → ∗ forms a covering family and so the stability axiom requires that G×_{{∗}}G = G × G be in R_{G}^{+}.

3.3. Example. Let C be any finite discrete G-set with more than one element and with trivial G-action, g: G → C any morphism, and consider the cover

{f_{i}: C_{i} → C| i ∈ I} ∈ K(C),

where C_{j} = C and f_{j}: C → C is the morphism mapping C to g(1), for some j ∈ I. Because the action is trivial, f_{j} is G-equivariant. There certainly exist covers of C of this form, since one could let f_{k} = id_{C}, for some k ≠ j in I, and then let the other f_{i} be any morphisms with codomain C. Hence, the stability axiom requires that

G ×_{C} C = {(γ, c)| g(γ) = f_{j}(c)} = {(γ, c)| γ · g(1) = g(1)} = G_{g(1)} × C = G × C

exists in R_{G}^{+}, which is impossible.

Thus, the stability axiom for a pretopology must be altered so that one still obtains a topology. We list the correct axioms for our situation below. They are taken from [MM94, pg. 156].

1. If f: C′ → C is an isomorphism, then {f: C′ → C} ∈ K(C).

2. (stability axiom) If {f_{i}: C_{i} → C| i ∈ I} ∈ K(C), then for any morphism g: D → C, there exists a cover {h_{j}: D_{j} → D| j ∈ J} ∈ K(D) such that for each j, g ∘h_{j} factors through some f_{i}.

3. (transitivity axiom) If {f_{i}: C_{i} → C| i ∈ I} ∈ K(C), and if for each i ∈ I there is a family {g_{ij}: D_{ij} → C_{i}| j ∈ I_{i}} ∈ K(C_{i}), then the family of composites

{f_{i} ∘ g_{ij}: D_{ij} → C| i ∈ I, j ∈ I_{i}}

is in K(C).

Proof of Theorem 3.1. It is clear that the pretopology K satisfies axiom (1) above. Also, it is easy to see that axiom (3) holds. Indeed, using the above notation, choose any c ∈ C. Then there is some c_{i} ∈ C_{i} for some i, such that f_{i}(c_{i}) = c. Similarly, there must be some d_{ij} ∈ D_{ij} for some j, such that g_{ij}(d_{ij}) = c_{i}. Hence, (f_{i} ∘ g_{ij})(d_{ij}) = f_{i}(c_{i}) = c, so that ∐_{i,j} D_{ij} → C is onto. This verifies (3). We verify (2) by considering five cases.

Case (1 ): Suppose that D and each of the C_{i} are finite sets in R_{G}^{+}. By Lemma 2.1, C must be a finite set. Consider the cover {π_{L}(i): D ×_{C} C_{i} → D| i ∈ I}, where each π_{L}(i) equals π_{L} and each g ∘ π_{L}(i) factors through f_{i} via the canonical map π_{R}. Now, for any d ∈ D, there exists some l ∈ I such that g(d) = f_{l}(c_{l}), for some c_{l} ∈ C_{l}. Thus, (d, c_{l}) ∈ D ×_{C} C_{l}, so that ∐_{i∈I} D ×_{C} C_{i} → D maps (d, c_{l}) to d and is therefore an epimorphism. This shows that {π_{L}(i)| i ∈ I} is in K(D).

Case (2 ): Suppose that D = G and that each C_{i} is a finite set in R_{G}^{+}. By Lemma 2.1, C is a finite set and we identify it with ∐_{i=1}^{n} G/U_{i}, where U_{i} = G_{x_{i}}, the stabilizer in G of x_{i}. The map g is determined by g(1) = δU_{k}, for some δ ∈ G and some stabilizer U_{k}. Since ∐_{i∈I} C_{i} → C is onto and im(g) = G/U_{k}, there exists some c_{l} ∈ C_{l}, for some l, such that f_{l}(c_{l}) = U_{k}. Since C_{l} is a finite set, we can identify c_{l} with some μG_{z}, where μ ∈ G and G_{z} is the stabilizer of some element z ∈ C_{l}.

Define the cover to be {λ: G → G}, where λ(γ) = γδ^{−1}. Define α_{l}: G → C_{l} to be the G-equivariant map given by 1 ↦ μG_{z}. By Lemma 2.5, α_{l} is continuous and is a morphism in R_{G}^{+}. Since λ is a homeomorphism, the cover {λ} is in K(D). Now,

(g ∘ λ)(1) = g(δ^{−1}) = δ^{−1} · g(1) = U_{k} = μ · f_{l}(G_{z}) = μ · f_{l}(μ^{−1} · α_{l}(1)) = (f_{l} ∘ α_{l})(1).

This shows that g ∘ λ factors through f_{l} via α_{l}.

Case (3 ): Suppose that not all the C_{i} are finite sets and that D = G. Also, assume that C = G, so that C_{i} = G, for all i ∈ I. Choose any k ∈ I, let α_{k} = id_{G}, and define λ: G → G to be multiplication on the right by f_{k}(1)g(1)^{−1}. Then g ∘ λ = f_{k} ∘ id_{G}, since

(g ∘ λ)(1) = g(f_{k}(1)g(1)^{−1}) = f_{k}(1)g(1)^{−1} · g(1) = f_{k}(1) = (f_{k} ∘ α_{k})(1),

so that the stability axiom is satisfied by letting the covering family be {λ}.

Case (4 ): Suppose that not all the C_{i} are finite sets, D = G, and C is a finite set. With C as in Lemma 2.4, let g(1) = δU_{k} ∈ C, as in Case (2). Then there exists some l such that f_{l}(c_{l}) = U_{k}, for some c_{l} ∈ C_{l}. Now we consider two subcases.

Case (4a ): Suppose that C_{l} is a finite set in R_{G}^{+}. Just as in Case (2), we construct maps λ and α_{l}, so that g ∘ λ factors through f_{l} via α_{l} and {λ} ∈ K(D).

Case (4b ): Suppose that C_{l} = G. By G-equivariance, f_{l}(1) = c_{l}^{−1}U_{k}. Then define λ: G → G by 1 ↦ δ^{−1} and α_{l}: G → G by 1 ↦ c_{l}. Then g ∘ λ factors through f_{l} via α_{l}, since

(g ∘ λ)(1) = g(δ^{−1}) = δ^{−1} · g(1) = U_{k} = f_{l}(c_{l}) = (f_{l} ∘ α_{l})(1).

Thus, {λ} ∈ K(D), since λ is a homeomorphism. This completes Case (4).

Now we consider the final possibility, Case (5 ): suppose that not all of the C_{i} are finite sets and suppose that D is a finite set. This implies that C is a finite set. This case is more difficult than the others because it combines several of the previous constructions and the desired cover consists of more than one morphism. For each d ∈ D, we make a choice of some i(d) ∈ I and some c_{d} ∈ C_{i(d)}, such that c_{d} is in the preimage of g(d) under ∐_{i∈I} C_{i} → C. Then write D = D_{df} ∐ D_{G}, where D_{df} is the set of all d such that C_{i(d)} is a finite set, and D_{G} is the set of all d such that C_{i(d)} = G. Now consider the cover {h_{d}: D_{d} → D| d ∈ D}, where

D_{d} = \begin{cases}D ×_{C} C_{i(d)} & if d ∈ D_{df}, \\ G & if d ∈ D_{G} \end{cases}

and h_{d} is defined in the following paragraph.

If d ∈ D_{df}, then set h_{d} = π_{L} and let α_{d}: D ×_{C} C_{i(d)} → C_{i(d)} be the canonical map π_{R}; it is clear that the required square commutes. Now suppose d ∈ D_{G}. Then there exists c_{d} ∈ C_{i(d)} = G, such that g(d) = f_{i(d)}(c_{d}). We write f_{i(d)}(1) = θU_{k} ∈ C, for some θ ∈ G and for some stabilizer U_{k}. Then we define α_{d}: G → C_{i(d)} = G by 1 ↦ θ^{−1}. Also, we define h_{d}: G → D by 1 ↦ (θ^{−1}c_{d}^{−1}) · d. Lemma 2.5 shows that h_{d} is a morphism in R_{G}^{+}. Then g ∘ h_{d} = f_{i(d)} ∘ α_{d}, as desired, since

(g ∘ h_{d})(1) = g((θ^{−1}c_{d}^{−1}) · d) = (θ^{−1}c_{d}^{−1}) · g(d)
= (θ^{−1}c_{d}^{−1}) · f_{i(d)}(c_{d}) = f_{i(d)}(θ^{−1}) = (f_{i(d)} ∘ α_{d})(1).

The only remaining detail is to show that {h_{d}| d ∈ D} ∈ K(D); that is, we must show that ϕ: ∐_{d∈D} D_{d} → D is an epimorphism. Let d be any element in D. Suppose d ∈ D_{df}, with c_{d} ∈ C_{i(d)}, such that f_{i(d)}(c_{d}) = g(d). Then (d, c_{d}) ∈ D ×_{C} C_{i(d)} and ϕ(d, c_{d}) = π_{L}(d, c_{d}) = d. Now suppose d ∈ D_{G}. With c_{d} ∈ G and θ as above, c_{d}θ ∈ D_{d} = G and ϕ(c_{d}θ) = h_{d}(c_{d}θ) = (c_{d}θ) · h_{d}(1) = d. Therefore, ϕ is an epimorphism.

4. A second pretopology on R_{G}^{+}

In this section, we obtain a second pretopology on R_{G}^{+} that generates a strictly coarser topology than that given by K. We use L to denote the pretopology of epimorphic covers for the site G−Sets_{df}.

Let M be the function on R_{G}^{+} defined by

M(C) = \begin{cases}L(C) & if C ∈ G−Sets_{df}, \\ \{R | R ⊂ Hom_{R_{G}^{+}}(G, G); R ≠ ∅\} & if C = G. \end{cases}

Thus, M consists exactly of the covers from the sites G (the groupoid) and G−Sets_{df}. Now we show that M makes R_{G}^{+} a site.

4.1. Theorem. For any profinite group G, the category R_{G}^{+} is a site with pretopology M.

Proof. As in the proof of Theorem 3.1, since R_{G}^{+} does not have enough pullbacks, we use the alternative set of axioms for a pretopology stated just before the proof of Theorem 3.1.

Now suppose that f: C′ → C is an isomorphism in R_{G}^{+}. If C is in G−Sets_{df}, then C′ is too. Since f is a morphism in the category G−Sets_{df}, {f} ∈ L(C) = M(C). If C = G, then C′ = G and it is clear that {f} ∈ M(G). This verifies the first axiom.

Axiom three is also true, since, using the notation of its statement above (just before the proof of Theorem 3.1), all the C_{i} and D_{ij} are in only G−Sets_{df} or in only groupoid G, according to whether C is in G−Sets_{df} or G, respectively.

In the notation of Axiom 2 (as stated in Section 3), suppose that C = G, so that each C_{i} is equal to G. If D = ∅, then it is easy to see that the stability axiom is satisfied by using the cover {∅ → ∅} ∈ M(∅). Now suppose that D = G. Then case (3) in the proof of Theorem 3.1 shows that Axiom 2 holds.

Now suppose C ∈ G−Sets_{df} and consider any {f_{i}: C_{i} → C| i ∈ I} ∈ L(C). All the C_{i} must be finite sets, and, if g: D → C is any morphism in R_{G}^{+}, then D is either a finite set or G. If D is a finite set, then case (1) of the proof of Theorem 3.1 gives the required cover, since G−Sets_{df} is closed under pullbacks. Finally, if D = G, then case (2) of the proof of Theorem 3.1 gives a satisfactory cover of G.

As stated at the beginning of this section, it is not hard to see that M generates a strictly coarser topology than K. For example, for K, the trivial one-element G-set {∗} has a covering sieve of the form

({G → {∗}}) = {G → {∗}, ∅ → {∗}},

which is clearly not a covering sieve for M.

5. Model category structures on presheaves of spectra on the site R_{G}^{+}

By Theorems 3.1 and 4.1, we know how to make the category R_{G}^{+} into a site via pretopologies K and M, respectively. Thus, in this section, we consider model category structures on the category of presheaves of spectra on the site (R_{G}^{+}, K), where K is any pretopology on R_{G}^{+}.

Let Ab be the category of abelian groups, and let Spt denote the model category of Bousfield-Friedlander spectra of pointed simplicial sets. We refer to the objects of Spt as simply “spectra.” Also, we let PreSpt(R_{G}^{+}, K) be the category of presheaves of spectra on the site (R_{G}^{+}, K).

Now we give some interesting examples of objects in PreSpt(R_{G}^{+}, K).

5.1. Example. In the Introduction, we saw that the Devinatz-Hopkins functor F is an object in PreSpt(R_{G_{n}}^{+}, K).

For the next example, if X is a spectrum, then, for each k ≥ 0, we let X_{k} be the kth pointed simplicial set constituting X, and, for each l ≥ 0, X_{k,l} is the pointed set of l-simplices of X_{k}.

5.2. Example. Let X be a discrete G-spectrum (see [Dav06b] for a definition of this term), so that each X_{k,l} is a pointed discrete G-set. If C ∈ R_{G}^{+}, then let Hom_{G}(C, X) be the spectrum, such that

Hom_{G}(C, X)_{k} = Hom_{G}(C, X_{k}),

where

Hom_{G}(C, X)_{k,l} = Hom_{G}(C, X_{k})_{l} = Hom_{G}(C, X_{k,l}).

Above, the set X_{k,l} is given the discrete topology, since it is naturally a discrete G-set. Then Hom_{G}(−, X) is an object in PreSpt(R_{G}^{+}, K). It is easy to see that if U is an open subgroup of G, then Hom_{G}(G/U, X) ≅ X^{U}, the U-fixed point spectrum of X. Also, Hom_{G}(G, X) ≅ X.

Now we recall [Dav06a, Lemma 3.1], since this result (and its corollary below (see [Dav06a, (3.3)])) will be helpful in our next example. We note that this result is only a slight extension of [Jar97, Remark 6.26]: if U is normal in G, then the lemma below is an immediate consequence of Jardine’s remark.

5.3. Lemma. Let X be a discrete G-spectrum. Also, let f: X → X_{f,G} be a trivial cofibration, such that X_{f,G} is fibrant, where all this takes place in the model category of discrete G-spectra (see [Dav06b]). If U is an open subgroup of G, then X_{f,G} is fibrant in the model category of discrete U-spectra.

5.4. Corollary. Let X and U be as in the preceding lemma. Then X^{hU} = (X_{f,G})^{U}.

Proof. Let f be as in the above lemma. Since f is G-equivariant, it is U-equivariant. Also, since f is a trivial cofibration in the model category of discrete G-spectra, it is a trivial cofibration in the model category of spectra. The preceding two facts imply that f is a trivial cofibration in the model category of discrete U-spectra. By the lemma, X_{f,G} is fibrant in this model category. Thus, X^{hU} = (X_{f,G})^{U}.

5.5. Example. Let X be a discrete G-spectrum. Then Hom_{G}(−, X_{f,G}) is a presheaf in PreSpt(R_{G}^{+}, K). In particular, notice that

Hom_{G}(G/U, X_{f,G}) ≅ (X_{f,G})^{U} = X^{hU}

and

Hom_{G}(G, X_{f,G}) ≅ X_{f,G} ≃ X.

5.6. Example. For any unfamiliar concepts in this example, we refer the reader to [Dav06b]. Let Z = holim_{i} Z_{i} be a continuous G-spectrum, so that {Z_{i}}_{i≥0} is a tower of discrete G-spectra, such that each Z_{i} is a fibrant spectrum. Then

P (−) = holim_{i} Hom_{G}(−, (Z_{i})_{f,G}) ∈ PreSpt(R_{G}^{+}, K),

where

P (G/U) ≅ holim_{i}((Z_{i})_{f,G})^{U} = holim_{i}(Z_{i})^{hU} = Z^{hU}

and

P (G) ≅ holim_{i}(Z_{i})_{f,G} ≃ Z.

The above examples show that the category PreSpt(R_{G}^{+}, K) gives an efficient way of organizing homotopy fixed point data for discrete and continuous G-spectra. Also, notice that when X is a discrete G-spectrum, Example 5.5 shows that the presheaf Hom_{G}(−, X_{f,G}) has X as its G-sections (up to weak equivalence), whereas a theorem is required to show if and when the presheaf of spectra Hom_{G}(−, X_{f,G}) on the site G−Sets_{df} has X as sections (up to weak equivalence). This observation is a minor organizational advantage of the category PreSpt(R_{G}^{+}, K) over the category of presheaves of spectra on G−Sets_{df}.

Now we equip PreSpt(R_{G}^{+}, K) with a model category structure M_{c}; let PreSpt^{c}(R_{G}^{+}, K) be the resulting model category. Notice that since Examples 5.1, 5.2, 5.5, and 5.6 are independent of the model category structure, PreSpt^{c}(R_{G}^{+}, K) is always useful for organizing homotopy fixed points “in the G-world.” However, it is natural to wonder if there is a particular model category PreSpt^{c}(R_{G}^{+}, K) that can provide some additional utility with regard to homotopy fixed points. Though we have not succeeded in finding a good answer to this question, we are able to make a few observations in the following discussion.

First, we briefly review the more important model structures M_{c} that are available. In making the recollections that follow, we were aided by the helpful exposition in [CHSW08, pp. 560–561], and any facts or results in the remainder of this section that are stated without citation or justification can be found in these pages.

Our review begins with some definitions. Let f: P → Q be a morphism of presheaves of spectra on R_{G}^{+}. Then f is an objectwise weak equivalence (objectwise cofibration, objectwise fibration) if f(C) is a weak equivalence (cofibration, fibration) of spectra, for all C ∈ R_{G}^{+}. Then M_{in}, the injective model structure, is the model category structure where the weak equivalences and cofibrations are the objectwise weak equivalences and the objectwise cofibrations, respectively (see [Jar87], [Jar97, Remark 2.36]).

5.7. Definition. Let P: (R_{G}^{+})^{op} → Spt be a presheaf of spectra. Then, for each n ∈ ℤ,

π_{n}(P): (R_{G}^{+})^{op} → Ab, C ↦ π_{n}(P (C)),

is a presheaf of abelian groups. Then the associated sheaf π̃_{n}(P) of abelian groups is the sheafification of π_{n}(P).

We say that the morphism f of presheaves is a local stable equivalence if the induced map π̃_{n}(P) → π̃_{n}(Q) of sheaves is an isomorphism, for all n ∈ ℤ. Then M_{L(in)}, the local injective model structure, is defined by taking the weak equivalences and cofibrations to be the local stable equivalences and objectwise cofibrations, respectively (see [Jar87], [Jar97, Theorem 2.34]). In M_{L(in)}, a fibration is referred to as a global fibration.

Additionally, M_{pr}, the projective model structure, is given by taking the weak equivalences and fibrations to be the objectwise weak equivalences and objectwise fibrations, respectively. In M_{pr}, each cofibration (that is, each projective cofibration) is an objectwise cofibration. Also, there is M_{L(pr)}, the local projective model structure, in which the weak equivalences are the local stable equivalences and the cofibrations are exactly the projective cofibrations (see [Bla01, Section 1] for an explanation of the terminology).

As alluded to in Lemma 5.3, homotopy fixed point spectra for profinite group actions are defined by using the model category structure on the category of discrete G-spectra that is specified by defining a morphism f to be a weak equivalence (cofibration) if it is a weak equivalence (cofibration) of spectra. As explained in [Dav06b, Section 3], this model category structure is obtained from the local injective model structure on the category PreSpt(G−Sets_{df}) (the category of presheaves of spectra on the canonical site G−Sets_{df}), in which the weak equivalences are the local stable equivalences and the cofibrations are defined objectwise.

Now, since G−Sets_{df} is a full subcategory of R_{G}^{+}, we can make the following definition.

5.8. Definition. If P is a presheaf of spectra in PreSpt(R_{G}^{+}, K), let P_{df} be the presheaf in PreSpt(G−Sets_{df}) that is obtained by the composition

(G−Sets_{df})^{op} ↪^{i} (R_{G}^{+})^{op} →^{P} Spt,

where i is the inclusion. Similarly, if f is a morphism in PreSpt(R_{G}^{+}, K), let f_{df} be the induced morphism in PreSpt(G−Sets_{df}). Thus, there is the restriction functor

i_{∗}: PreSpt(R_{G}^{+}, K) → PreSpt(G−Sets_{df}), P ↦ i_{∗}(P) = P_{df}.

Now we make a few observations about the above model category structures and i_{∗}. As before, let f be a morphism in PreSpt(R_{G}^{+}, K). It is easy to see that if f is a cofibration in M_{in} or M_{L(in)}, then i_{∗}(f) is a cofibration in PreSpt(G−Sets_{df}) (equipped with the local injective model category structure described above). Similarly, since any projective cofibration is an objectwise cofibration, if f is a cofibration in M_{pr} or M_{L(pr)}, then i_{∗}(f) is a cofibration in PreSpt(G−Sets_{df}).

If f is an objectwise weak equivalence, then i_{∗}(f) is also an objectwise weak equivalence. Thus, for each integer n and for any object C in G − Sets_{df}, π_{n}(f(C)) is an isomorphism, so that the map π_{n}(i_{∗}(f)) of presheaves is an isomorphism. Hence, the associated map π̃_{n}(i_{∗}(f)) of sheaves on G−Sets_{df} is an isomorphism, implying that i_{∗}(f) is a local stable equivalence. Thus, if f is a weak equivalence in M_{in} or M_{pr}, then i_{∗}(f) is a weak equivalence in PreSpt(G−Sets_{df}).

Now suppose that f is a weak equivalence between fibrant presheaves in the model category PreSpt^{L(in)}(R_{G}^{+}, K). Since a global fibration is a fibration in M_{L(pr)}, f is also a weak equivalence between fibrant presheaves in PreSpt^{L(pr)}(R_{G}^{+}, K). It follows (see [CHSW08, pg. 561]) that f is an objectwise weak equivalence. Thus, we can conclude that if f is a weak equivalence between globally fibrant presheaves in PreSpt^{L(in)}(R_{G}^{+}, K), then i_{∗}(f) is a weak equivalence in PreSpt(G−Sets_{df}).

Given the last statement, it is natural to wonder if i_{∗} preserves globally fibrant objects: if P is a fibrant presheaf in PreSpt^{L(in)}(R_{G}^{+}, K), one can ask if i_{∗}(P) is fibrant. Similarly, one can ask if i_{∗} preserves all global fibrations. Since global fibrations in PreSpt^{L(in)}(R_{G}^{+}, K) are defined to be the maps with the right lifting property with respect to all trivial cofibrations, the definition of local stable equivalence implies that to answer these questions, one needs a good understanding of sheaves on the site (R_{G}^{+}, K). In particular, these matters could be considered when K is K or M. However, we have not succeeded in carrying out such an analysis.

References

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[CHSW08] G. Cortiñas, C. Haesemeyer, M. Schlichting, and C. Weibel, Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. 167 (2008), no. 2, 549–573.

[Dav06a] Daniel G. Davis, Iterated homotopy fixed points for the Lubin-Tate spectrum (with an appendix by Daniel G. Davis and Ben Wieland), Topology Appl. 156 (2009), no. 17, 2881–2898.

[Dav06b] Daniel G. Davis, Homotopy fixed points for L_{K(n)}(E_{n} ∧ X) using the continuous action, J. Pure Appl. Algebra 206 (2006), no. 3, 322–354.

[Dem70] Michel Demazure, Topologies et faisceaux, Schémas en groupes. SGA 3, I: Propriétés générales des schémas en groupes, Springer-Verlag, Berlin, 1970, pp. xv+564.

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[DH04] Ethan S. Devinatz and Michael J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004), no. 1, 1–47.

[EKMM97] A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole.

[Jar87] J. F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math. 39 (1987), no. 3, 733–747.

[Jar97] J. F. Jardine, Generalized étale cohomology theories, Birkhäuser Verlag, Basel, 1997.

[MM94] Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic, Springer-Verlag, New York, 1994, A first introduction to topos theory, Corrected reprint of the 1992 edition.

[Rud98] Yuli B. Rudyak, On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998, With a foreword by Haynes Miller.

University of Louisiana Mathematics Department 1403 Johnston Street Maxim Doucet Hall, Room 217 Lafayette, LA 70504

Email: dgdavis@louisiana.edu

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