THE SITE R+G FOR A PROFINITE GROUP G
DANIEL G. DAVIS
Abstract. Let G be a nonfinite profinite group and let G  Setsdfbe th*
*e canonical
site of finite discrete Gsets. Then the category R+G, defined by Devina*
*tz and Hopkins,
is the category obtained by considering G  Setsdftogether with the prof*
*inite Gspace
G itself, with morphisms being continuous Gequivariant maps. We show th*
*at R+Gis a
site when equipped with the pretopology of epimorphic covers. Also, we e*
*xplain why the
associated topology on R+Gis not subcanonical, and hence, not canonical.*
* We note that,
since R+Gis a site, there is automatically a model category structure on*
* the category of
presheaves of spectra on the site. Finally, we point out that such presh*
*eaves of spectra
are a nice way of organizing the data that is obtained by taking the hom*
*otopy fixed
points of a continuous Gspectrum with respect to the open subgroups of *
*G.
1. Introduction
Let G be a profinite group that is not a finite group. Let R+Gbe the category w*
*ith objects
all finite discrete left Gsets together with the left Gspace G. The morphisms*
* of R+Gare
the continuous Gequivariant maps. Since G is not finite, the object G in R+Gi*
*s very
different in character from all the other objects of R+G. In this paper, we sho*
*w that R+Gis
a site when equipped with the pretopology of epimorphic covers.
As far as the author knows, the category R+Gis first defined and used in the*
* paper
[Devinatz and Hopkins, 2004], by Ethan Devinatz and Mike Hopkins. Let Gn be the*
* profi
nite group Sn o Gal(Fpn=Fp), where Sn is the nth Morava stabilizer group. In [D*
*evinatz
and Hopkins, 2004, Theorem 1], Devinatz and Hopkins construct a contravariant f*
*unctor
 that is, a presheaf 
F: (R+Gn)op ! (E1 )K(n),
to the category (E1 )K(n) of K(n)local commutative Salgebras (see [Elmendorf *
*et. al.,
1997]), where K(n) is the nth Morava Ktheory (see [Rudyak, 1998, Chapter 9] fo*
*r an
exposition of K(n)). The functor F has the properties that, if U is an open sub*
*group of Gn,
then F(Gn=U) = EdhUn, and F(Gn) = En, where En is the nth LubinTate spectrum (*
*for
salient facts about En and its importance in homotopy theory, see [Devinatz and*
* Hopkins,
1995, Introduction]), and EdhUnis a spectrum that behaves like the Uhomotopy f*
*ixed point
spectrum of En with respect to the continuous Uaction. Since Hom R+G (Gn, Gn) *
*~=Gn,
n
functoriality implies that Gn acts on En by maps of commutative Salgebras. In *
*Section
_____________
2000 Mathematics Subject Classification: 55P42, 55U35, 18B25.
Key words and phrases: site, profinite group, finite discrete Gsets, preshe*
*aves of spectra, LubinTate
spectrum, continuous Gspectrum.
Oc Daniel G. Davis, 2006. Permission to copy for private use granted.
1
2
5, we will give several related examples of presheaves of spectra that illustra*
*te the utility
of the category R+G.
The pretopology of epimorphic covers on a small category`C is the pretopolog*
*y K given
by all covering families {fi:Ci ! C i 2 I} such that OE: i2ICi ! C is onto, *
*where
Ci, C 2 C, fi2 Mor C(Ci, C), and I is some indexing set. (Of course, one must p*
*rove that
these covering families actually give a pretopology on C.) We note that we do n*
*ot require
that OE be a morphism in C; for our purposes, C = R+Gand we only require that O*
*E be an
epimorphism in the category of all Gsets (so that OE does not have`to be conti*
*nuous).
This assumption is important for our work, since, for example, G G is not in *
*R+G.
The pretopology K is a familiar one. For example, for a profinite group G, *
*K is
the standard basis used for the site G  Setsdfof finite discrete Gsets ([Jard*
*ine, 1997,
pg. 206]). However, there is an important difference between R+Gand G  Setsd*
*f: the
latter category is closed under pullbacks, but it is easy to see that R+Gdoes n*
*ot have
all pullbacks (this point will be discussed later). But in a category with pul*
*lbacks, the
canonical topology, the finest topology in which every representable presheaf i*
*s a sheaf,
is given by all covering families of universal effective epimorphisms (see Expo*
*se IV, 4.3
of [Demazure, 1970]). This implies that G  Setsdfis a site with the canonical *
*topology
when equipped with pretopology K. However, due to the lack of sufficient pullba*
*cks, we
cannot conclude that K gives R+Gthe canonical topology. In fact, we will show *
*that K
does not generate the canonical topology, since K does not yield a subcanonical*
* topology.
Note that R+Gis built out of the two subcategories GSets dfand the groupoid*
* G.
Since each of these categories is a site via K (for G, this is verified in Lemm*
*a 2 below), it
is natural to think that R+Gis 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. More gener*
*ally, there
is the category PreSpt(R+G) of presheaves of spectra on R+G. Furthermore, since*
* R+Gis a
site, the work of Jardine (e.g., [Jardine, 1987], [Jardine, 1997]) implies that*
* PreSpt(R+G)
is a model category. We recall the definition of this model category in Section*
* 5.
In [Davis, 2006], the author showed that, given a continuous Gspectrum Z, t*
*hen, for
any open subgroup U of G, there is a homotopy fixed point spectrum ZhU , define*
*d with
respect to the continuous action of U on Z. In Examples 5.7 and 5.8, we see tha*
*t there is a
presheaf that organizes in a functorial way the following data: Z, ZhU for all *
*U open in G,
and the maps between these spectra that are induced by continuous Gequivariant*
* maps
between the Gspaces G and G=U. Thus, PreSpt(R+G) is a natural category within *
*which
to work with continuous Gspectra. It is our hope that the model category struc*
*ture 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.
Acknowledgements. When I first tried to make R+Ga site, and was focusing on an
abstract way of doing this, Todd Trimble helped me get started by suggesting th*
*at I
extend K to all of R+G. Also, I thank him for pointing out Lemma 2.1. I thank*
* Paul
Goerss for discussions about this material. Also, I appreciate various conversa*
*tions with
Christian Haesemeyer about this work.
3
2. Preliminaries
Before we prove our main results, we first collect some easy facts which will b*
*e 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= GSets dfand there is nothing to prove.)
2.1. Lemma. Let f :C ! G be any morphism in R+Gwith C 6= ?. Then C = G.
Proof. Choose any c 2 C and let f(c) = fl. Choose any ffi 2 G. Then
ffi = (ffifl1)fl = (ffifl1) . f(c) = f((ffifl1) . c),
by the Gequivariance of f. Thus, f is onto and im (f) = 1, so that C cannot*
* be a
finite set. *
* 
2.2. Lemma. For a topological group G, let G be the groupoid with the single ob*
*ject G and
morphisms the Gequivariant maps G ! G given by right multiplication by some el*
*ement
of G. Then G is a site with the pretopology K of epimorphic covers.
f g
Proof. Any diagram G ! G G, where f and g are given by multiplication by fl*
* and
ffi, respectively, can be completed to a commutative square
f0
G ____//_G
g0 g
fflfflffflffl
G ____//_G,
where f0 and g0 are given by multiplication by ffi1 and fl1, respectively. Th*
*is property
suffices to show that G is a site with the atomic topology, in which every siev*
*e is a covering
sieve if and only if it is nonempty. It is easy to see that the only nonempty s*
*ieve of G is
Mor G(G, G) itself. Thus, the only covering sieve of G is the maximal sieve. Si*
*nce every
morphism of G is a homeomorphism, in the pretopology K, the collection of cover*
*s 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 Gequivariance, f is *
*the map
given by multiplication by f(1) on the right. As mentioned earlier, we have
2.3. Lemma. The category GSetsdf, a full subcategory of R+G, is closed under p*
*ullbacks.
Proof. The pullback of a diagram in GSetsdfis formed simply by regarding the d*
*iagram
as being in the category TG of discrete Gsets. The category TG is closed under*
* pullbacks,
as explained in [Mac Lane and Moerdijk, 1994, pg. 31]. *
* 
4
We recall the following useful result and its proof.
` n __
2.4. Lemma. Let X be any finite set in R+G. We write X = i=1xi, the disjoint *
*union
of`all the distinct orbits __xi, with each xi a representative. Then X is home*
*omorphic to
n
i=1G=Ui, where Ui= Gxiis the stabilizer in G of xi.
Proof. Let f :G=Ui ! __xibe given by f(flUi) = fl . xi. Since X is a discrete G*
*set, the
stabilizer Uiis an open subgroup of G with finite index, so that G=Uiis a finit*
*e 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 flUi= ffiUi. Then fl1ffi 2 Ui, so that (fl1ffi) . xi= (f*
*l1) . (ffi . xi) = xi.
Thus, fl . xi = ffi . xi and f is welldefined. Assume that fl . xi = ffi . xi.*
* Then fl1ffi 2 Gxi
so that f is a monomorphism. *
* 
2.5. Lemma. Let X be a finite discrete Gset in R+G and let _ :G ! X be any G
equivariant function. Then _ is a morphism in R+G.
` n
Proof. As in Lemma 2.4, we identify X with i=1G=Ui. Since _ is Gequivariant *
*and
_(fl) = fl . _(1), _ is determined by _(1). Let _(1) = ffiUj for some ffi 2 G a*
*nd some j.
Then for any fl in G,
flUj = (flffi1ffi)Uj = (flffi1) . _(1) = _(flffi1),
so that im _ = G=Uj. Since X is discrete, _ is continuous, if, for any x 2 X, _*
*1(x) is
open in G. It suffices, by the identification, to let x = flUj, for any fl 2 G.*
* Then
_1(flUj)= {i 2 G _(i) = flUj} = {i 2 G i . (ffiUj) = flUj}
= {i 2 G ffi1i1fl 2 Uj} = flUjffi1.
Since Uj is open and multiplication on the left or the right is always a homeom*
*orphism
in a topological group, we see that _1(x) is an open set in G. *
* 
3. The proof of the main theorem
With these lemmas in hand, we are ready for
3.1. Theorem. For any profinite group G, the category R+Gequipped with the pret*
*opology
K of epimorphic covers is a small site.
Before proving the theorem, we first make some remarks about pullbacks in R+*
*Gand
how this affects our proof. In a category C with sufficient pullbacks, to prov*
*e that a
pretopology is given by a function K, which assigns to each object C a collecti*
*on K(C) of
families of morphisms with codomain C, one must prove the stability axiom, whic*
*h says
the following: if {fi:Ci ! C i 2 I} 2 K(C), then for any morphism g :D ! C, the
family of pullbacks
{ssL :D xC Ci! D i 2 I} 2 K(D).
Let us examine what this axiom would require of R+G.
5
3.2. Example. The map G ! * forms a covering family and so the stability axiom
requires that Gx{*}G = G x G be in R+G.
3.3. Example. Let C be any finite discrete Gset with more than one element and*
* with
trivial Gaction, g :G ! C any morphism, and consider the cover
{fi:Ci! C i 2 I} 2 K(C),
where Cj = C and fj: C ! C is the morphism mapping C to g(1), for some j 2 I.
Because the action is trivial, fj is Gequivariant. There certainly exist cover*
*s of C of this
form, since one could let fk = idC, for some k 6= j in I, and then let the othe*
*r fi be any
morphisms with codomain C. Then the stability axiom requires that G xC C exist*
*s in
R+G, but this is impossible, since
G xC C = {(fl, c) g(fl) = fj(c)} = {(fl, c) fl . g(1) = g(1)} = Gg(1)x C *
*= G x C.
Thus, the stability axiom for a pretopology must be altered so that one stil*
*l obtains
a topology. We list the correct axioms for our situation below. They are take*
*n from
[Mac Lane and Moerdijk, 1994, Exercise 3, pg. 156].
1.If f :C0! C is an isomorphism, then {f :C0! C} 2 K(C).
2.(stability axiom) If {fi:Ci! C i 2 I} 2 K(C), then for any morphism g :D *
*! C,
there exists a cover {hj: Dj ! D j 2 J} 2 K(D) such that for each j, g O *
*hj factors
through some fi.
3.(transitivity axiom) If {fi:Ci! C i 2 I} 2 K(C), and if for each i 2 I th*
*ere is a
family {gij:Dij! Ci j 2 Ii} 2 K(Ci), then the family of composites
{fiO gij:Dij! C i 2 I, j 2 Ii}
is in K(C).
Proof of Theorem 3.1. It is clear that the pretopology of epimorphic covers sat*
*isfies
axiom (1) above. Also, it is easy to see that axiom (3) holds. Indeed, using *
*the above
notation, choose any c 2 C. Then there is some ci2 Ci for some i, such that fi(*
*ci) = c.
Similarly, there must be some dij 2 Dij`for some j, such that gij(dij) = ci. H*
*ence,
(fiO gij)(dij) = fi(ci) = c, so that i,jDij! C is onto. This verifies (3). We*
* verify (2)
by considering five cases.
Case (1 ): Suppose that D and each of the Ci are finite sets in R+G. By Lemm*
*a 2.1, C
must be a finite set. Consider the cover
{ssL(i): D xC Ci! D i 2 I},
where ssL(i) is the obvious map and g O ssL(i) factors through fi via the canon*
*ical map
ssR(i). Now choose any d 2 D and let g(d) = c 2 C. Then there exists some i suc*
*h that
6
`
fi(ci) = c for ci2 Ci. Thus, (d, ci) 2 D xC Ci, so that ID xC Ci! D maps (d, *
*ci) to d
and is therefore an epimorphism. This shows that {ssL(i)} is in K(D).
Case (2 ): Suppose that D = G and that each`Ciis a finite set in R+G. By Lem*
*ma 2.1,
C is a finite set and we identify it with ni=1G=Ui, where Ui= Gxi, the stabil*
*izer of xiin
G.`The map g is determined by g(1) = ffiUk for some ffi 2 G and some stabilizer*
* Uk. Since
ICi ! C is onto and im(g) = G=Uk , there exists some cl 2 Cl such that fl(cl)*
* = Uk.
Since Cl is a finite set, we can identify cl with some ~Gz, where ~ 2 G and Gz *
*is the
stabilizer of some element z 2 Cl.
Then define the cover to be {~: G ! G}, where ~(fl) = flffi1. Define ffl:G *
*! Cl to
be the Gequivariant map given by 1 7! ~Gz. By Lemma 2.5, ffl is continuous and*
* is a
morphism in R+G. Since ~ is a homeomorphism, the cover {~} is in K(D). Now,
(g O ~)(1) = g(ffi1) = ffi1 . g(1) = Uk = ~ . fl(Gz) = ~ . fl(~1 . ffl(1)*
*) = (flO ffl)(1).
This shows that g O ~ factors through flvia ffl.
Case (3 ): Suppose not all the Ci are finite sets and that D = G. Also, assu*
*me that
C = G. This implies that Ci = G for all i 2 I. Choose any k 2 I, let ffk = id*
*G, and
define ~: G ! G to be multiplication on the right by fk(1)g(1)1. Then the diag*
*ram
idG
G ____//_G
~ fk
fflfflfflfflg
G ____//_G
is commutative, since
(g O ~)(1) = g(fk(1)g(1)1) = fk(1)g(1)1 . g(1) = fk(1) = (fk O ffk)(1*
*).
Thus, g O ~ factors through fk via ffk, so that the stability axiom is verified*
* by letting the
covering family be {~}.
Case (4 ): Suppose that not all the Ci are finite sets, D = G, and C is a fi*
*nite set.
With C as in Lemma 2.4, let g(1) = ffiUk 2 C, as in Case (2). Then there exists*
* some l
such that fl(cl) = Uk, for some cl2 Cl. Now we consider two subcases.
Case (4a ): Suppose that Cl is a finite set in R+G. Just as in Case (2), we*
* construct
maps ~ and ffl, so that g O ~ factors through flvia ffland {~} 2 K(D).
Case (4b ): Suppose that Cl = G. By Gequivariance, fl(1) = c1lUk. Then *
*define
~: G ! G by 1 7! ffi1 and ffl:G ! G by 1 7! cl. Then g O ~ factors through fl *
*via ffl,
since
(g O ~)(1) = g(ffi1) = ffi1 . g(1) = Uk = fl(cl) = (flO ffl)(1).
Thus, the cover {~}, as a homeomorphism, is in K(D). This completes Case (4).
Now we consider the final possibility, Case (5 ): suppose that not all of t*
*he Ci are
finite sets and suppose that D is a finite set. This implies that C is a finite*
* set. This case
7
is more difficult than the others because the cover consists of more than one m*
*orphism
and it combines the previous constructions. For each d 2 D, we make`a choice of*
* some
cl2 Cl for`some l, such that clis in the preimage of g(d) under Ci ! C. Then *
*write
D = Ddf DG , where Ddfis the set of all d such that the corresponding Clis in *
*a finite
set, and DG is the set of all d`such that the corresponding Cl = G. Now consid*
*er the
cover {hd: Dd ! D d 2 D = Ddf DG }, where
(
D xC Cd if d 2 Ddf,
Dd =
G if d 2 DG.
If d 2 Ddf, then hd = ssL and ffd: D xC Cd ! Cd is the canonical map ssR; it is*
* clear that
the required square commutes. Now suppose d 2 DG . Then there exists cl2 Cl= G *
*for
some l, such that g(d) = fl(cl). We write fl(1) = `Uk 2 C for some ` 2 G and fo*
*r some
stabilizer Uk. Then we define ffd: G ! Cl= G by 1 7! `1. Also, we define hd: G*
* ! D
by 1 7! (`1c1l) . d. Lemma 2.5 shows that hd is a morphism in R+G. Then we ha*
*ve the
required commutative diagram
ffd
G _____//G
hd fl
fflfflfflfflg
D ____//_C,
since
(g O hd)(1)= g((`1c1l) . d) = (`1c1l) . g(d)
= (`1c1l) . fl(cl) = fl(`1) = (flO ffd)(1).
The`only remaining detail is to show that {hd} 2 K(D); that is, we must show*
* that
OE: D Dd ! D is an epimorphism. Let d be any element in D. Suppose d 2 Ddf.
Then, using our choice above, there exists some cl2 Cl, a finite set for some l*
*, such that
fl(cl) = g(d). Then (d, cl) 2 D xC Cl and OE(d, cl) = ssL(d, cl) = d. Now suppo*
*se d 2 DG .
With cland ` as above, cl` 2 Dd = G and OE(cl`) = hd(cl`) = (cl`) . hd(1) = d. *
*Therefore,
OE is an epimorphism. *
* 
4. The site R+G does not have the canonical topology
Now that we have established that R+Gis a site with pretopology K, we begin wor*
*king to
show that, contrary to what typically happens with this pretopology, it does no*
*t give the
canonical topology. We start with a definition.
4.1. Definition. If T is some collection of morphisms with codomain C, where C *
*is an
object in the category C, then (T ) denotes the sieve generated by T . Thus,
(T ) = {f O g f 2 T, dom (f) = cod(g)}.
8
4.2. Lemma. Let K be a pretopology on a category C. Let J be the Grothendieck t*
*opology
generated by K. Then for any C 2 C, J(C) consists exactly of all (R) [ (T ) su*
*ch that
R 2 K(C) and T is some collection of morphisms with codomain C.
Proof. Let S be a covering sieve of C. Then there exists some R 2 K(C) such th*
*at
R S. We will prove that S = (R) [ (S), verifying the forward inclusion. To *
*prove
equality it suffices to show that (R) [ (S) S. If f 2 (R), f = g O h for some*
* g 2 R and
some h with dom (g) = cod(h). Since g 2 S, f 2 S. Similarly, if f 2 (S), then f*
* 2 S.
Now consider any family of morphisms (R) [ (T ) as described in the statement o*
*f the
lemma. Since R (R) [ (T ), (R) [ (T ) 2 J(C) if it is a sieve. Since (R) an*
*d (T ) are
sieves, it is clear that (R) [ (T ) is also a sieve. *
* 
This result is useful for understanding the topology of a site, when the sit*
*e is defined
in terms of a pretopology. For example, GSets dfis a site by the pretopology *
*K and
its category of sheaves of sets is equivalent to the category of sheaves on the*
* site S(G)
consisting of quotients of G by open subgroups (the morphisms are the Gequivar*
*iant
maps), where S(G) is given the atomic topology (see [Mac Lane and Moerdijk, 199*
*4,
Chapter 3, Section 9]). Thus, one might ask if GSets dfalso has the atomic top*
*ology.
However, the lemma allows us to see that K generates a topology`that is coarser*
* than
the atomic topology. To see this, let X = G=U and Y = G=U G=U, where U is
a proper open subgroup of G. (Since G is an infinite profinite group, the cano*
*nical
way of writing G as an inverse limit guarantees the existence of such a U.) We*
* define
f :X ! Y by f(U) = U, where U lives in the factor on the left; f is the left in*
*clusion.
Now consider`the sieve S = ({f}). Clearly, S does not contain an epimorphic cov*
*er, since
im ( g2S(dom (g)) ! Y ) = G=U. The lemma indicates that every sieve of GSetsd*
*fmust
contain an epimorphic cover, so that S is not a sieve for Y in the topology gen*
*erated by
K.
Now we consider the site R+Gwith the pretopology K of epimorphic covers. We*
* use
Hom G(X, Y ) to denote continuous Gequivariant maps between continuous Gsets*
* X and
Y . Recall that a presheaf of sets P on a site (C, J) is a sheaf, if for each o*
*bject C 2 C and
each covering sieve S 2 J(C), the diagram
Q _p__//_Q
P (C)__e_//_f2SP (dom (f))____//_P (dom (g))
a
is an equalizer of sets, where the second product is over all f, g, with f 2 S,*
* dom (f) =
cod(g). Here, e is the map e(x) = {P (f)(x)}f, p is given by
{xf}f 7! {xfg}f,g,
and a is given by
{xf}f 7! {P (g)(xf)}f,g= {xf O g}f,g.
Recall that a representable presheaf of R+Gis any presheaf which, up to isom*
*orphism,
has the form of Hom G(, C) for some C 2 R+G. Also, the Yoneda embedding
+)op
R+G! Sets(RG , C 7! Hom G(, C)
9
*
* +)op
is a full and faithful functor, so that one can identify C with an object of Se*
*ts(RG . We
now consider which objects of R+Gyield sheaves of sets on R+G.
Noting that the empty set is a discrete Gset, we have
4.3. Lemma. The presheaf Hom G (, ?) is a sheaf of sets on the site R+G.
Proof. Let o : ? ! X denote the vacuous map, for any X 2 R+G. Since o : ? ! ? is
vacuously an epimorphism, {o} is the unique covering sieve for ?. Let C = ?. Th*
*en the
desired equalizer diagram has the form
__p_//_
Hom G(?, ?) = {o} _e__//_{o}__//_{o}.
a
It is clear that this is an equalizer diagram.
Now let C be a nonempty finite set in G  Setsdf. Let S be any covering sie*
*ve of
C. There must exist a morphism in S with domain equal to a nonempty object in R*
*+G.
Therefore, since ? x Z = ? for any space Z, we have
__p__//
Hom G(C, ?) = ?__e__//?____//?.
a
Since the equalizer must exist and the vacuous map o : ? ! ? is the unique map *
*with
codomain ?, this must be an equalizer diagram.
Finally, letting C = G, we get
Q __p_//_
Hom G(G, ?) = ? __e_//_f2HomG(G,G)? = ? ____//_?.
a
Again, this is an equalizer diagram. *
* 
To prove the next theorem, we need the following lemma.
4.4. Lemma. If G is a compact topological group, U an open subgroup of G, and X*
* 6= ?
a finite discrete Gset, then
Hom G(G=U, X) ~={x 2 X U < Gx},
where Gx is the stabilizer of x in G.
Proof. Let f : G=U ! X. It is clear that f is Gequivariant if and only if it i*
*s completely
determined by f(U) in the obvious way. Since U is an open subgroup, it has fini*
*te index
in G, so that G=U is a discrete space. Thus, any Gequivariant map G=U ! X is
continuous. The key is that f is welldefined if and only if U < Gf(U). To see *
*this, first
assume that f is welldefined; let fl 2 U. Then flU = U, so that fl .f(U) = f(f*
*lU) = f(U).
Hence, fl 2 Gf(U)and U < Gf(U). Now suppose that U < Gf(U)and take any flU = ff*
*iU.
This implies that fl1ffi 2 U and hence, in Gf(U). Thus, (fl1ffi) . f(U) = f(*
*U), so that
fl . f(U) = ffi . f(U). Equivariance gives f(flU) = f(ffiU) and f is welldefin*
*ed. Thus,
Hom G(G=U, X) ~={f(U) 2 X U < Gf(U)}.

10
Henceforth, let J denote the topology of R+Ggenerated by K.
4.5. Theorem. Let X be any object in R+Gthat is not a finite discrete trivial G*
*set, where
G is an infinite profinite group. Then the presheaf Hom G (, X) is not a sheaf*
* of sets on
the site R+G.
Proof. Suppose Hom G (, X) is a sheaf of sets on the site R+G. The equalizer c*
*ondition
says that for every object C 2 R+Gand for every covering sieve S 2 J (C),
Hom G (C, X) ~={{hf}f hfg = hf O g, f, g, f 2 S, dom (f) = cod(g)},
where for f 2 S, hf 2 Hom G(dom (f), X). We will construct an example of some C*
* and
S such that this sheaf condition fails to`be true with X as above.
Let C 2 GSetsdf; we identify C with ni=1G=Ui, where each Uiis an open sub*
*group
of G. For each i, define fi:G ! C by 1 7! Ui. Thus, im(fi) = G=Ui and {fi} is*
* an
epimorphic cover of C. The preceding lemma tells us that S = ({fi}) is a coveri*
*ng sieve
of C. For this S, we will examine the sheaf condition. Let S = S0[ S00, where S*
*0= {fi}
and S00is the complement of S0in S. Thus, every k 2 S00has the form k = fiO g f*
*or some
g with dom (fi) = cod(g). Then
{{hf}f hfg= hf O g, f, g, f 2 S, dom (f) = cod(g)}
= {{hfi}fix {hk}k2S00 hfg = hf O g, f, g, f 2 S, dom (f) = cod(g)}
= {{hfi}fix {hfiO g}fiOg2S00 hfg = hf O g, f, g, f 2 S, dom (f) = c*
*od(g)}
= {{hfi}fix {hfiO g}fiOg2S00 hfi2Hom G(G, X), fi2S0, g, dom (fi) =*
* cod(g)}.
We verify the last equality. Suppose hfiis any morphism in Hom G(G, X). Now tak*
*e any f
and g with f 2 S and dom (f) = cod(g). If f = fi2 S0, then hfOg = hfiOg = hfiOg*
*= hfOg,
by construction. Now suppose f 2 S00. Then f = fiO k for some k :G ! G. Thus,
hfg = hfiO(kOg)= (hfiO k) O g = hfiOkO g = hf O g.
Since hfiO g is determined by hfiand fiO g, we see that the set
{{hfi}fix {hfiO g}fiOg2S00 hfi2 Hom G(G, X), fi2 S0, g, dom (fi) = cod(*
*g)}
is isomorphic to the set
{{hfi}fi hfi2 Hom G(G, X), fi2 S0} = Hom G(G, X)n,
where Hom G (G, X)n is the nfold Cartesian product of Hom G (G, X). Now, ther*
*e is an
isomorphism Hom G (G, X)n ~=Xn. Therefore, for Hom G (, X) to be a sheaf, it m*
*ust be
that Hom G(C, X) ~=Xn for every C 2 G  Setsdf. If X = G and C 6= ? is in G  S*
*etsdf,
then Hom G (C, G) = ?, whereas, since C 1, n 1 and Xn = Gn. Thus, Hom G (*
*, G)
is not a sheaf.
11
Now we consider X 6= G and assume that Hom G(C, X) ~=Xn for every C 2 GSets*
*df.
This implies that
` n
Xn ~=Hom G(C, X) ~=Hom G ( i=1G=Ui, X)
~=Q ni=1HomG(G=Ui, X) ~=Q ni=1{x 2 X Ui< Gx} Xn.
Therefore, it must be that {x 2 X Ui< Gx} =`X, for all i = 1, ..., n. Thus, Ui*
*< Gx for
all x 2 X and each i. Now let us write X ~= mj=1G=Gxj, where each xj is a repr*
*esentative
from a distinct orbit of X. Let C be a trivial Gset so that every stabilizer o*
*f c 2 C in G
is equal to G. This implies that G < Gxj for all j. Thus, each Gxj = G. This in*
*dicates
that X must be a trivial Gset. This contradiction shows that every X violates *
*the sheaf
condition for some C and S. *
* 
This result immediately yields
4.6. Corollary. For an infinite profinite group G, the site R+Gwith the pretopo*
*logy K
of epimorphic covers is not subcanonical.
Proof. There exists a proper open subgroup U of G satisfying`[G : U] > 1. Thus,*
* the
representable presheaves Hom G (, G) and Hom G (, ni=1G=U), for any n 1, *
*are not
sheaves. *
* 
Since a canonical topology is, by definition, subcanonical, we obtain
4.7. Corollary. For an infinite profinite group G, the site R+G, with the preto*
*pology K,
is not canonical.
The next result is an elementary fact about profinite groups that helps us u*
*nderstand
"how often" representable presheaves fail to be sheaves in R+Gand what such "fa*
*iling"
presheaves can look like, based on what we know from Theorem 4.5.
4.8. Lemma. If G is an infinite profinite group, then G contains an infinite nu*
*mber of
distinct proper open subgroups.
Proof. We have already seen that G has at least one proper open subgroup. Suppo*
*se
that G has only a finite number of distinct proper open subgroups. Then G has a*
* finite
numberTof distinct proper open normal subgroups N1, ..., Nk. Since G is profini*
*te, N =
k
i=1Ni = {1}. Because N is an open subgroup with finite index, it has uncount*
*able
order. This contradiction gives the conclusion. *
* 
4.9. Remark. Since any topology finer than J would contain the covering sieve (*
*{fi})
that was the key to Theorem 4.5, no topology finer than J can be subcanonical.
5. Presheaves of spectra on the site R+G
Let Ab be the category of abelian groups, and let Spt denote the model categor*
*y of
BousfieldFriedlander spectra of pointed simplicial sets. We refer to the obje*
*cts of Spt
12
as simply "spectra." Now that R+Gis a site, we can consider the category PreSpt*
*(R+G) of
presheaves of spectra on the site R+G. By applying the work of Jardine ([Jardin*
*e, 1987],
[Jardine, 1997, Section 2.3]), PreSpt (R+G) is a model category. We recall the*
* critical
definitions that give the model category structure and then we state Jardine's *
*result,
when it is applied to R+G.
5.1. Definition. Let P :(R+G)op ! Spt be a presheaf of spectra. Then, for each *
*n 2 Z,
ssn(P ): (R+G)op ! Ab , C 7! ssn(P (C)),
is a presheaf of abelian groups. Then the associated sheaf "ssn(P ) of abelian *
*groups is the
sheafification of ssn(P ).
Let f :P ! Q be a morphism of presheaves of spectra on R+G. Then f is a weak
equivalence if the induced map "ssn(P ) ! "ssn(Q) of sheaves is an isomorphism,*
* for all
n 2 Z. The map f is a cofibration if f(C) is a cofibration of spectra, for all *
*C 2 R+G. Also,
f is a global fibration if f has the right lifting property with respect to all*
* morphisms
which are weak equivalences and cofibrations.
5.2. Theorem. [Jardine, 1997, Theorem 2.34] The category PreSpt(R+G), together *
*with
the classes of weak equivalences, cofibrations, and global fibrations, is a mod*
*el category.
Now we give some interesting examples of presheaves of spectra on the site R*
*+G.
5.3. Example. In the Introduction, we saw that the DevinatzHopkins functor F i*
*s an
example of an object in PreSpt(R+Gn).
For the next example, if X is a spectrum, then, for each k 0, we let Xk be*
* the
kth pointed simplicial set constituting X, and, for each l 0, Xk,lis the poin*
*ted set of
lsimplices of Xk.
5.4. Example. Let X be a discrete Gspectrum (see [Davis, 2006] for a definitio*
*n of this
term), so that each Xk,lis a pointed discrete Gset. If C 2 R+G, then let Hom G*
* (C, X) be
the spectrum, such that
Hom G(C, X)k = Hom G(C, Xk),
where
Hom G(C, X)k,l= Hom G(C, Xk)l= Hom G(C, Xk,l).
Above, the set Xk,lis given the discrete topology, since it is naturally a disc*
*rete Gset.
Then Hom G (, X) is an object in PreSpt(R+G). It is easy to see that if U is *
*an open
subgroup of G, then Hom G (G=U, X) ~= XU , the Ufixed point spectrum of X. Al*
*so,
Hom G(G, X) ~=X.
Now we recall part of [Behrens and Davis, 2005, Proposition 3.3.1], since th*
*is result
(and its corollary) will be helpful in our next example. We note that this resu*
*lt is only
a slight extension of [Jardine, 1997, Remark 6.26]: if U is normal in G, then t*
*he lemma
below is an immediate consequence of Jardine's remark.
13
5.5. Lemma. Let X be a discrete Gspectrum. Also, let f :X ! Xf,Gbe a trivial c*
*ofibra
tion, such that Xf,Gis fibrant, where all this takes place in the model categor*
*y of discrete
Gspectra (see [Davis, 2006]). If U is an open subgroup of G, then Xf,G is fibr*
*ant in the
model category of discrete Uspectra.
5.6. Corollary. Let X and U be as in the preceding lemma. Then XhU = (Xf,G)U.
Proof. Let f be as in the above lemma. Since f is Gequivariant, it is Uequiva*
*riant.
Also, since f is a trivial cofibration in the model category of discrete Gspec*
*tra, it is a
trivial cofibration in the model category of spectra. The preceding two facts i*
*mply that
f is a trivial cofibration in the model category of discrete Uspectra. By the *
*lemma, Xf,G
is fibrant in this model category. Thus, XhU = (Xf,G)U. *
* 
5.7. Example. Let X be a discrete Gspectrum. Then Hom G (, Xf,G) is a preshea*
*f in
PreSpt(R+G). In particular, notice that
Hom G(G=U, Xf,G) ~=(Xf,G)U = XhU
and
Hom G(G, Xf,G) ~=Xf,G' X.
5.8. Example. For any unfamiliar concepts in this example, we refer the reader *
*to [Davis,
2006]. Let Z = holimiZibe a continuous Gspectrum, so that {Zi}i 0is a tower of*
* discrete
Gspectra, such that each Zi is a fibrant spectrum. Then
P () = holimHom G(, (Zi)f,G) 2 PreSpt(R+G),
i
where
P (G=U) ~=holim((Zi)f,G)U = holim(Zi)hU = ZhU
i i
and
P (G) ~=holim(Zi)f,G' Z.
i
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Department of Mathematics
Wesleyan University
265 Church St.
Middletown, CT 064590128
Email: dgdavis@wesleyan.edu