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November 25, 2010

Locally Constant Sheaves

Posted by Mike Shulman

Urs and I recently got into a discussion about the correct definition of “locally constant ∞-sheaf.” In trying to sort it out, I realized that I don’t even know the right definition of a locally constant 1-sheaf, in general! Specifically, there are two definitions of “locally constant sheaf” which are the same when the base space (or, more generally, topos) is locally connected—but otherwise they’re not the same. Can anyone tell me which is the right definition in general, and why?

Here they are. Let XX be a topological space, for maximum familiarity and concreteness.

Definition 1. A sheaf FF on XX is locally constant if there exists an open cover (U i)(U_i) of XX such that F| U iF|_{U_i} is (isomorphic to) a constant sheaf for each ii.

(Recall that for a set AA, the constant sheaf ΔA\Delta A on a space UU is the sheafification of the constant presheaf. Regarded as a local homeomorphism to UU, it is the space U×AU\times A. Its sections over an open set VUV\subseteq U are the locally constant functions VAV\to A, i.e. functions which are continuous when AA has the discrete topology.)

Definition 2. Let Core(Set)Core(Set) denote the groupoid of sets and isomorphisms, and Core(Set *)Core(Set_\ast) the groupoid of pointed sets and pointed isomorphisms, with forgetful functor Core(Set *)pCore(Set)Core(Set_\ast) \xrightarrow{p} Core(Set). A locally constant sheaf is a pullback of Δ(p)\Delta(p) along a map *ΔCore(Set)\ast\to \Delta Core(Set) in the 2-category Sh (2,1)(X)Sh_{(2,1)}(X) of stacks of groupoids on XX, where Δ\Delta denotes the constant stack functor GpdSh (2,1)(X)Gpd \to Sh_{(2,1)}(X).

This second definition takes a bit of unraveling; we need to think about what a constant stack looks like. Of course, it is a stackification of a constant prestack (= groupoid-valued presheaf, in this case). If GG is a groupoid, then a global section of the constant stack ΔG\Delta G is determined by

  1. an open cover (U i)(U_i) of XX,
  2. for each ii, an object a iGa_i\in G,
  3. for each i,ji,j, a locally constant function g ij:U iU jHom G(a i,a j)g_{i j}\colon U_i \cap U_j \to Hom_G(a_i,a_j),
  4. such that g jk(x)g ij(x)=g ik(x)g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x) for any i,j,ki,j,k and any xU iU jU kx\in U_i \cap U_j\cap U_k.

Thus, a global section of ΔCore(Set)\Delta Core(Set) is given by

  1. an open cover (U i)(U_i) of XX,
  2. for each ii, a set A iA_i,
  3. for each i,ji,j, a locally constant function g ij:U iU jIso(A i,A j)g_{i j}\colon U_i \cap U_j \to Iso(A_i,A_j),
  4. such that g jk(x)g ij(x)=g ik(x)g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x) for any i,j,ki,j,k and any xU iU jU kx\in U_i \cap U_j\cap U_k.

Note that we can regard the g ijg_{i j} as sections over U iU jU_i \cap U_j of the constant sheaf ΔIso(A i,A j)\Delta Iso(A_i,A_j). The corresponding sheaf on XX is obtained by gluing the constant sheaves ΔA i\Delta A_i on U iU_i together along the isomorphisms g ijg_{i j}. Note that if XX is connected, then all the sets A iA_i must be isomorphic, so we could equivalently give just a single set AA together with locally constant functions g ij:U iU jAut(A)g_{i j}\colon U_i \cap U_j \to Aut(A). In other words, when XX is connected, a locally constant sheaf on XX (according to definition 2) is a principal Aut(A)Aut(A)-bundle over XX, for some (discrete) set AA. Equivalently, we can call this a ΔAut(A)\Delta Aut(A)-torsor in the topos Sh(X)Sh(X).

Now let’s go back to definition 1. We again have an open cover (U i)(U_i), and for each ii we have F| U iΔA iF|_{U_i} \cong \Delta A_i, for some set A iA_i, so this looks very similar. But now instead of the gluing data g ij:U iU jIso(A i,A j)g_{i j}\colon U_i \cap U_j \to Iso(A_i,A_j), we are simply given a sheaf FF on all of XX which restricts to ΔA i\Delta A_i on each U iU_i. Since the assignment USh(U)U\mapsto Sh(U) is a stack on XX, to be given such an FF is equivalent to being given isomorphisms h ij:ΔA i| U iU jΔA j| U iU jh_{i j}\colon \Delta A_i |_{U_i\cap U_j} \cong \Delta A_j |_{U_i\cap U_j} which satisfy the cocycle condition. And again, if XX is connected, then we could equivalently give ourselves just one set AA and automorphisms h ijAut Sh(X)(ΔA| U iU j)h_{i j} \in Aut_{Sh(X)}(\Delta A|_{U_i\cap U_j}).

Thus, clearly the difference between definitions 1 and 2 is the difference between an automorphism of a constant sheaf and a global section of a constant sheaf of automorphisms. What is that difference? Well, since Δ\Delta preserves cartesian products, and SetSet and Sh(X)Sh(X) are both cartesian closed, for any sets AA and BB there is a canonical map Δ(B A)(ΔB) ΔA\Delta(B^A) \to (\Delta B)^{\Delta A}. If this map is always an isomorphism, we call Δ\Delta a cartesian closed functor. And the “object of isomorphisms” Iso(A,B)Iso(A,B) in any cartesian closed category can be constructed as an equalizer of a pair of maps A B×B AA A×B BA^B \times B^A \rightrightarrows A^A \times B^B, and since Δ\Delta also preserves equalizers, if it is a cartesian closed functor, then we have ΔIso(A,B)Iso(ΔA,ΔB)\Delta Iso(A,B) \cong Iso(\Delta A,\Delta B) for any sets AA and BB.

Now it’s a classical fact that Δ\Delta is a cartesian closed functor if and only if XX is locally connected. Thus, in this case we have ΔIso(A,B)Iso(ΔA,ΔB)\Delta Iso(A,B) \cong Iso(\Delta A,\Delta B), which implies that the two definitions of “locally constant sheaf” are the same. (One could also give a more abstract argument.) However, if XX is not locally connected, then we can have ΔIso(A,B)Iso(ΔA,ΔB)\Delta Iso(A,B) \ncong Iso(\Delta A,\Delta B).

For instance, suppose XX is the one-point compactification of \mathbb{N}. Then for any set AA, a global section of ΔA\Delta A is a function A\mathbb{N}\to A which is eventually constant. In particular, a global section of Aut(A)Aut(A) is an eventually constant sequence of automorphisms of AA.

On the other hand, since ΔA\Delta A is also the coproduct of AA copies of the terminal object, a map ΔAΔA\Delta A \to \Delta A is just an AA-indexed family of eventually constant functions f x:Af_x\colon \mathbb{N}\to A. In particular, an automorphism of ΔA\Delta A is such a family such that for every nn, the function xf x(n)x \mapsto f_x(n) is an automorphism of AA, as is the limit function f f^\infty defined by f (x)=f x(n)f^\infty(x) = f_x(n) for sufficiently large nn. (See below.)

So the difference between a global section of ΔAut(A)\Delta Aut(A) and an automorphism of ΔA\Delta A is that the former is an eventually constant sequence of automorphisms, whereas the latter is a sequence of automorphisms such that for each xAx\in A, the sequence of images of xx is eventually constant. Evidently the latter is strictly more general if AA is infinite, so we cannot have ΔAut(A)Aut(ΔA)\Delta Aut(A) \cong Aut(\Delta A).

When I first posted this, I thought I could use this to construct an example of a sheaf on the closed topologist’s sine curve satisfying definition 1 but not definition 2. However, now I don’t think my example quite works any more. But maybe some more similar but more complicated example would work.

So are the two definitions the same? If not, which is the right definition of a locally constant sheaf on a non-locally-connected space, and why?

Posted at November 25, 2010 12:54 AM UTC

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Re: Locally Constant Sheaves

Nice post. Haven’t had a chance to think about it yet. None the less, a question:

Isn’t there also a definition in terms of maps from the fundamental groupoid? If so, which of the two definitions you list is it equivalent to?

Posted by: Eugene Lerman on November 25, 2010 5:08 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Isn’t there also a definition in terms of maps from the fundamental groupoid? If so, which of the two definitions you list is it equivalent to?

Ah, indeed, you refer to

Definition 3. A locally constant sheaf on XX is a functor Π 1(X)Core(Set)\Pi_1(X) \to Core(Set).

For general XX, and with the “usual” definition of Π 1(X)\Pi_1(X) in terms of paths, I don’t think this is equivalent to either of the first two definitions. For instance, if XX is something like the Warsaw circle, then Π 1(X)\Pi_1(X) is trivial when defined in terms of paths, but there are nonconstant locally constant sheaves on XX according to either of the first two definitions.

Of course, the problem is that for spaces like this, the usual definition of the fundamental groupoid isn’t good enough. And getting a notion of fundamental groupoid which makes Definition 3 equivalent to (one of) the others is one of the main reasons for redefining it.

On the one hand, if XX is locally 1-connected, then it has a fundamental groupoid Π 1(X)\Pi_1(X) which makes Definition 3 equivalent to both of the first two (which are the same in that case, since locally 1-connected implies locally 0-connected, i.e. locally connected). But at least from a higher-topos-theory point of view, the equivalence with Definition 2 is more direct: XX being locally 1-connected means that the constant stack functor Δ:GpdSh (2,1)(X)\Delta\colon Gpd \to Sh_{(2,1)}(X) has a left adjoint Π\Pi, so that global sections of ΔCore(Set)\Delta Core(Set) are equivalent to groupoid maps Π(*)Core(Set)\Pi(\ast)\to Core(Set). Thus defining Π 1(X)Π(*)\Pi_1(X)\coloneqq \Pi(\ast) works.

If XX is not locally 1-connected, then we can’t hope for an ordinary fundamental groupoid, but we can hope for a fundamental progroupoid. From a higher-topos-theory point of view, this progroupoid is most naturally characterized by the functor GpdGpdGpd \to Gpd that it corepresents. Thus, the equivalence with Definition 2 is also more direct in this case.

However, all the papers and books that I can remember ever reading about fundamental progroupoids give Definition 1 of locally constant sheaves—but also usually restrict to the case when XX is locally connected (at least eventually). So it could be that a historically myopic focus on locally connected spaces has obscured the difference between Definitions 1 and 2 and that for non-locally-connected ones, Definition 2 is actually better. But I hesitate to accuse all these authors (from Grothendieck on down, as far as I know) of myopia!

Posted by: Mike Shulman on November 25, 2010 5:57 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Hmm, two corrections to the original version of this post. Firstly, I wrote, about sheaves on the one-point compactification of \mathbb{N}:

… since ΔA\Delta A is also the coproduct of AA copies of the terminal object, a map ΔAΔA\Delta A \to \Delta A is just an AA-indexed family of eventually constant functions f x:Af_x\colon \mathbb{N}\to A. In particular, an automorphism of ΔA\Delta A is such a family such that for every nn, the function xf x(n)x \mapsto f_x(n) is an automorphism of AA.

But I neglected to add the requirement that the corresponding function f f^\infty (the endomorphism of the stalk over the point at infinity), which is defined by f (x)=f x(n)f^\infty(x) = f_x(n) for sufficiently large nn, must also be an automorphism. I guess insofar as I thought about it at all, I assumed that would follow automatically, but in fact it doesn’t. For instance, we could take A=A=\mathbb{N} and f x(n)={x+1 ifx<n 0 ifx=n x ifx>n f_x(n) = \begin{cases} x+1 &\quad if x\lt n\\ 0 &\quad if x=n\\ x &\quad if x\gt n \end{cases} Then each xf x(n)x\mapsto f_x(n) is an automorphism, and each sequence (f x(n)) n(f_x(n))_{n\in \mathbb{N}} is eventually constant at x+1x+1, so f (x)=x+1f^\infty(x) = x+1, which is not an automorphism.

I think saying that f f^\infty is an automorphism is equivalent to saying that the sequence of inverses of the functions f n(x)=f x(n)f^n(x) = f_x(n) is also pointwise eventually constant.

Secondly, I also wrote:

let XX be the closed topologist’s sine curve: the graph of sin(1/x)\sin(1/x) for x>0x\gt 0, together with {0}×[1,1]\{0\}\times [-1,1], with the subspace topology from 2\mathbb{R}^2. Let U=X(×(12,1])U = X \cap (\mathbb{R}\times (-\frac{1}{2} ,1]) and V=X(×[1,12))V = X \cap (\mathbb{R}\times [-1,\frac{1}{2} )). Then UU and VV are open subsets of XX with UV=XU \cup V = X, so to specify a sheaf on XX satisfying definition 1, it suffices to give a set AA and an automorphism of ΔA\Delta A over UVU\cap V.

However, I claim that UVU\cap V is homeomorphic to an open interval times the one-point compactification of \mathbb{N}. (Draw a picture.) Thus, if AA is an infinite set, we can pick an automorphism of Δ UVA\Delta_{U\cap V} A which is constant in the open interval direction, and does not come from a global section of Δ UVAut(A)\Delta_{U\cap V} Aut(A). The sheaf on XX resulting from gluing Δ UA\Delta_U A to Δ VA\Delta_V A along this automorphism will satisfy definition 1 but not definition 2.

But I neglected to think about whether that sheaf on the closed topologist’s sine curve might be isomorphic to one obtained as in Definition 2, even though it doesn’t look on the surface like one of that form.

Let’s see, if two sheaves FF and GG are obtained by gluing Δ UA\Delta_U A to Δ VA\Delta_V A along some pointwise-eventually-constant sequence of automorphisms f nAut(A)f_n\in Aut(A) and g nAut(A)g_n\in Aut(A), as above, then an isomorphism FGF\cong G would be given by automorphisms of Δ UA\Delta_U A and Δ VA\Delta_V A which commute with the ff’s and gg’s.

Since UU and VV also look like the one-point compactification of \mathbb{N}, with each point fattened up into a connected space, automorphisms of Δ UA\Delta_U A and Δ VA\Delta_V A must also be given by pointwise-eventually-constant sequences h n,k nAut(A)h_n,k_n \in Aut(A). Looking at how UU and VV match up with UVU\cap V, the commutation conditions should be something like h nf 2n=g 2nk nh_n f_{2n} = g_{2n} k_n and h nf 2n+1=g 2n+1k n+1h_n f_{2n+1} = g_{2n+1} k_{n+1}.

I think this means that any sheaf FF on XX satisfying Definition 1 is actually isomorphic to the constant sheaf Δ XA\Delta_X A. Of course taking G=Δ XAG=\Delta_X A means that g n=idg_n = \id for all nn. Now I claim that for any FF, we can define the hh’s and kk’s inductively as follows: let k 0=idk_0=\id, then we must have h 0=f 0 1h_0 = f_0^{-1}, then k 1=h 0f 1k_1 = h_0 f_1, then h 1=k 1f 2 1h_1 = k_1 f_2^{-1}, then k 2=h 1f 3k_2 = h_1 f_{3}, and so on. These hh’s and kk’s and their inverses are pointwise eventually constant because we have h n+1=h nf 2n+1f 2n+2 1h_{n+1} = h_n f_{2n+1} f_{2n+2}^{-1} and so on, and the ff’s and their inverses are pointwise eventually constant.

In retrospect, this is not surprising, since the space XX intuitively “has no loops,” so from the Π 1\Pi_1 perspective we wouldn’t expect it to have any interesting locally constant sheaves. Currently I feel like it ought to be possible to cook up some weirder non-locally-connected space on which we could define a sheaf satisfying Definition 1 but not Definition 2, though I haven’t yet thought of one. But maybe they really are the same….

Posted by: Mike Shulman on November 25, 2010 11:05 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

How about looking at pp-adic solenoids? These are not nice spaces, and key examples in shape theory. One might say the problem with the topologists sine curve is that it only has one problem point (or if you like, a small-ish subspace), away from which everything is hunky-dory. Solenoids are a bit more uniformly nasty.

Posted by: David Roberts on November 26, 2010 5:32 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

The dyadic solenoid does occur in various papers with some sort of fibring over it. (The papers tend to be in dynamical systems so I am by no means competent to comment on the constructions.) Of interest also may be the work of Alex Clark at Leicester. (See his homepage.) Directly he is looking at bundles over spaces with fibre a solenoid, which is the wrong way around, and I have not looked further yet. This is all in the context of smooth foliations. It may be worth checking out. (If I find anything I will post it on the forum.)

Posted by: Tim Porter on November 26, 2010 7:58 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Thanks for the suggestion, David. I think I’ve got an example on a simpler space, but if it also turns out not to work, I may take a look at solenoids.

First note that since S 1S^1 is connected and locally 1-connected with Π 1(S 1)=\Pi_1(S^1)=\mathbb{Z}, all the definitions agree over S 1S^1, and a locally constant sheaf over S 1S^1 is determined by a set AA equipped with an automorphism ff. Given such, we can construct the locally constant sheaf explicitly by writing S 1=UVS^1=U\cup V, where UU and VV are homeomorphic to open intervals and UVU\cap V is homeomorphic to a disjoint union of two open intervals. Then we glue together Δ UA\Delta_U A and Δ VA\Delta_V A along 1 A1_A and ff on the two components of UVU\cap V, respectively.

Now let X=S 1× X = S^1 \times \mathbb{N}_\infty, where \mathbb{N}_\infty is the one-point compactification of \mathbb{N}. Write U=U× U' = U\times \mathbb{N}_\infty and V=V× V' = V \times \mathbb{N}_\infty, and let (f n)Aut(Δ A)(f_n)\in Aut(\Delta_{\mathbb{N}_\infty} A) be an automorphism not coming from ΔAut(A)\Delta Aut(A), i.e. a pointwise-eventually-constant sequence of automorphisms of AA which is not globally eventually constant. Now UV=(UV)× U' \cap V' = (U\cap V) \times \mathbb{N}_\infty, so it is a disjoint union of two spaces that look like (0,1)× (0,1)\times \mathbb{N}_\infty. Then the automorphism (f n)(f_n) can be extended in the open interval direction to give an automorphism of ΔA\Delta A over (0,1)× (0,1)\times \mathbb{N}_\infty.

Let the sheaf FF be obtained by gluing Δ UA\Delta_{U'}A to Δ VA\Delta_{V'}A along 1 A1_A and (f n)(f_n) on the two copies of (0,1)× (0,1)\times \mathbb{N}_\infty that make up UVU'\cap V'. By construction, FF satisfies Definition 1; I claim that it is not isomorphic to anything constructed as in Definition 2.

To see this, suppose GG satisfies Definition 2. Then, in particular, XX is covered by open sets over which GG is isomorphic to constant sheaves. If GG is going to be isomorphic to FF, then these constant sheaves will all have to be constant at the set AA.

Now consider the subspace S 1×{}XS^1 \times \{\infty\} \subseteq X; since this is compact, it is covered by finitely many such opens, which we may take to be of the form W×[n,]W \times [n,\infty] for some open WS 1W\subset S^1 and nn\in \mathbb{N} (these being the basic open neighborhoods in XX of points in S 1×{}S^1\times \{\infty\}). Let mm be the maximum of this finite number of nn’s; thus GG is constant over each W i×[m,]W_i\times [m,\infty] for some finite open cover (W i)(W_i) of S 1S^1. We may assume that each W iW_i is homeomorphic to an open interval, as is each nonempty intersection W 1W 2W_1\cap W_2.

Since GG satisfies Definition 2, over each (W 1×[m,])(W 2×[m,])(W_1\times [m,\infty])\cap (W_2\times [m,\infty]), the constant sheaves ΔA\Delta A must be glued together along a section of ΔAut(A)\Delta Aut(A) over (W 1W 2)×[m,](W_1\cap W_2) \times [m,\infty]. Since W 1W 2W_1\cap W_2 is connected and [m,] [m,\infty]\cong \mathbb{N}_\infty, such a section is again just a globally eventually constant sequence of automorphisms of AA. Let k>mk\gt m be large enough that all of these sequences become constant after kk. Then using the cover (W i)(W_i) of S 1S^1 and the limit automorphisms over each intersection, we can construct a uniquely specified locally constant sheaf on S 1S^1, call it HH. As a locally constant sheaf on S 1S^1 with fiber AA, HH is determined by a single automorphism of AA.

Moreover, since all the gluing data is constant after kk, for any >k\ell\gt k the restriction of GG to S 1×{}S^1\times \{\ell\} must be isomorphic to HH. However, the sheaf FF does not satisfy this property for any locally constant sheaf HH on S 1S^1, since the restrictions of FF to S 1×{}S^1\times \{\ell\} are determined by the automorphisms (f n)(f_n). Since these are not eventually constant, they determine a sequence of locally constant sheaves on S 1S^1 which are also not eventually constant, even up to isomorphism.

Posted by: Mike Shulman on November 26, 2010 7:31 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

By the way, I think the difference between the two definitions becomes more serious as you go up in categorical dimension. The two notions of locally constant 1-sheaf agree for locally connected spaces, so analogously in general I think that the corresponding definitions of locally constant (n,1)-sheaf will agree only for locally (n-1)-connected spaces. In particular, for locally constant (∞,1)-sheaves, the definitions will agree only for locally contractible spaces.

Posted by: Mike Shulman on November 26, 2010 7:35 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I agree these two notions won’t agree in general. However, unless I missed something, definition 1 is really what we are looking for (if we want to relate the pro-groupoid associated to a topos with a notion of locally constant sheaf), despite the fact that it is natural to feel more secure with definition 2.

If ever one believes (as I do) that locally constant sheaves will form a full subcategory of the category of all sheaves, definition 1 will come back anyway: locally constant sheaves should form as stack, and the guys of definition 1 are exactly those which are locally like those of definition 2. The main point here is thus to understand how we may understand locally constant sheaves in the sense of definition 1 if we understand locally constant sheaves in the more restrictive sense of definition 2.

Let XX be a locally constant sheaf (in the sense of definition 1). Then there exists a (non necessarily unique) decomposition of the terminal sheaf 1=⨿ iU i1=\amalg_i U_i, and, for each ii a set A iA_i, such that X×U iX\times U_i is locally isomorphic to U i×ΔA iU_i\times \Delta A_i for each ii. By working over each U iU_i, we see that, for the study of XX, we may assume without loss of generality that there exists a set AA such that XX is locally isomorphic to ΔA\Delta A. Pick a covering V1V\to 1, and an isomorphism s:V×ΔAV×Xs:V\times\Delta A\to V\times X over VV. Denote by YY the sheaf of isomorphisms from ΔA\Delta A to XX, and write G=Aut(A)G=Aut(A) for the set of bijections from AA to itself. Write Y sY_s for the subsheaf of YY whose sections are those which are locally in the orbit of our given trivialization sY(V)s\in Y(V) under the natural action of GG (i.e. a section tY(U)t\in Y(U) belongs to Y s(U)Y_s(U) if there exists an epimorphism p:WU×Vp:W\to U\times V and an element gg of GG, such that t.gt.g and ss have the same image in Y(W)Y(W)). Then Y sY_s is a ΔG\Delta G-torsor, and Y s×XY_s\times X is constant over Y sY_s. In other words, we have the following nice property:

If XX is locally constant sheaf, there exists a decomposition of the terminal sheaf 1=⨿ iU i1=\amalg_i U_i such that, for any index ii, there exists a group G iG_i as well as a G iG_i-torsor Y iY_i over U iU_i with the property that X×Y iX\times Y_i is constant over Y iY_i.

Now, let us examine this story further (as before, in what follows, unless I specify it otherwise, I will consider only definition 1).

Let us start with the adjunction

π 1:TopoiPro(Groupoids):B\pi_1 : Topoi \rightleftarrows Pro(Groupoids) : B

where, for a pro-groupoid G=lim iG iG=lim_i G_i, BGBG is the 22-limit of the topoi BG i=G^ iBG_i=\hat{G}_i, and where π 1(T)\pi_1(T) is the pro-groupoid associated to TT (whatever it might be).

Define a 11-equivalence to be a (geometric) morphism of topoi f:TTf: T\to T' such that the associated map π 1(T)π 1(T)\pi_1(T)\to\pi_1(T') is invertible in Pro(Groupoids)Pro(Groupoids). This means that, for any groupoid GG, the map

Tors(T,G)=Hom(π 1(T),G)Hom(π 1(T),G)=Tors(T,G)Tors(T',G)=Hom(\pi_1(T'),G)\to Hom(\pi_1(T),G)=Tors(T,G)

is an equivalence of groupoids. Equivalently, this means that the morphism of topoi

π 1(T)π 1(T)\pi_1(T)\to \pi_1(T')

is an equivalence of topoi.

We then have a kind of toposic version of Quillen’s theorem A (with respect to 11-equivalences): consider a morphism of topoi f:TTf:T\to T' such that there exists a generating family GG of TT' satisfying the following assumption: for any XX in GG, the morphism T/f *XT/XT/f^*X\to T'/X is a 11-equivalence. Then ff itself is a 11-equivalence. (The proof is essentially the same as for classical theorem A: one shows that TT is the homotopy colimit of the T/f *XT/f^*X’s).

This implies the following

Proposition: Assume that f:TTf:T\to T' is a 11-equivalence. Then, for any locally constant sheaf XX on TT', the map T/f *XT/XT/f^*X\to T'/X is a 11-equivalence as well.

Proof: Using the toposic theorem A, we see that is is sufficient to treat the case where XX is constant. But if XX is the sheaf associated to the set AA, then, for any groupoid GG, we have:

Tors(T/X,G)=Tors(T,G) ATors(T,G) A=Tors(T/f *X,G).Tors(T'/X,G)=Tors(T',G)^A\simeq Tors(T,G)^A=Tors(T/f^*X,G)\, .

For a topos TT, write LC(T)LC(T) for the fullsubcategory of locally constant sheaves over TT. We get the

Corollary: If f:TTf:T\to T' is a 11-equivalence, then the inverse image functor f *:LC(T)LC(T)f^*:LC(T')\to LC(T) is an equivalence of categories.

Let us check that f *f^* is fully faithful on locally constant sheaves. Let XX and YY be locally constant sheaves over TT'. The property that the map

Hom T(X,Y)Hom T(f *X,f *Y)Hom_{T'}(X,Y)\to Hom_{T}(f^*X,f^*Y)

is bijective is local on XX (in the sense that we may replace XX by X×XX'\times X for any locally constant sheaf XX' such that X1X'\to 1 is an isomorphism). Therefore, we may assume that X×YX\times Y is constant over XX. By replacing TT' and TT by T/XT'/X and T/f *XT/f^*X respectively, we see that we may also assume that X=1X=1 (whence also that YY is constant). But the property of being a 11-equivalence certainly implies that f *f^* is fully faithful on constant sheaves. For the essential surjectivity, one uses the fact that, for any locally constant sheaf XX over TT, there exists a locally constant sheaf YY over TT' such that X×f *YX\times f^*Y is constant over f *Yf^*Y, and conclude with a descent argument.

In particular, if we apply the preceding corollary to the morphism TBπ 1(T)T\to B\pi_1(T), we get an equivalence of categories

LC(Bπ 1(T))LC(T).LC(B\pi_1(T))\simeq LC(T)\, .

Finally, note that, for any pro-groupoid GG, the topos BGBG admits a generating family which consists of locally constant sheaves. Therefore, LC(T)LC(T) (endowed with the canonical topology) is a site of definition for the topos Bπ 1(T)B\pi_1(T), and the full inclusion LC(T)TLC(T)\to T is a morphism of sites, from which we deduce that the category of sheaves on Bπ 1(T)B\pi_1(T) is canonically equivalent to the completion by colimits of the category LC(T)LC(T) in TT.

Posted by: Denis-Charles Cisinski on November 26, 2010 11:38 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I realized I didn’t formulate the toposic version of Quillen’s Theorem A with enough generality to make it work in the way I used it in my previous post: it is rather convenient to have a relative version as follows.

Let f:TTf:T\to T' and q:TSq:T'\to S be geometric morphisms of topoi, and write p=qfp=q f. Assume that there is a generating family GG of SS such that, for any XX in GG, the map

T/p *XT/q *XT/p^*X\to T'/q^*X

is a 11-equivalence, then the map ff is a 11-equivalence.

When I referred to the toposic theorem A in the proof of the proposition in my previous post, it was rather to this relative version for S=Bπ 1(T)S=B\pi_1(T').

Also, it seems that Galois theory for non-necessarily locally connected topoi has been worked out by Eduardo J. Dubuc in this paper

The fundamental progroupoid of a general topos, JPAA 212 (2008), 2479-2492.

(also available as arXiv:0706.1771).

As you can see, he works with locally constant sheaves in the sense of definition 1.

Posted by: Denis-Charles Cisinski on November 27, 2010 9:50 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Hooray! An expert showed up.

I haven’t digested most of your comments yet, but already in the second paragraph I think you hit the nail on the head regarding the intuitive difference between the two definitions.

If ever one believes (as I do) that locally constant sheaves will form a full subcategory of the category of all sheaves

In other words, is being “locally constant” a property of a sheaf, or structure (or even stuff) that can be given to it?

To make that even more explicit: the 2-category of 2-sheaves (= stacks) on any site (or even any 2-site) has a ternary factorization system which is analogous to the (faithful, full, essentially surjective) one on CatCat, except that everything is “localized” so we get (locally faithful, locally full, locally essentially surjective). A map F:MNF\colon M\to N of 2-sheaves is:

  • locally faithful if for any UU, any x,yM(U)x,y\in M(U) and α,βHom M(U)(x,y)\alpha,\beta \in Hom_{M(U)}(x,y), if F(α)=F(β)F(\alpha)=F(\beta), then there is a covering family (V iU)(V_i\to U) of UU such that α| V i=β| V i\alpha|_{V_i} = \beta|_{V_i} for all ii. (Since MM is a 2-sheaf, this is equivalent to α=β\alpha=\beta, so localizing faithfulness doesn’t change it.)
  • locally full if for any UU, any x,yM(U)x,y\in M(U), and any γHom N(U)(F(x),F(y))\gamma \in Hom_{N(U)}(F(x),F(y)), there exists a covering family (V iU)(V_i\to U) of UU and morphisms α iHom M(V i)(x| V i,y| V i)\alpha_i \in Hom_{M(V_i)}(x|_{V_i},y|_{V_i}) such that F(α i)=γ| V iF(\alpha_i) = \gamma|_{V_i} for all ii. (This is different from plain fullness, but localizing “full and faithful” also doesn’t change it, i.e. a map of 2-sheaves is (locally) faithful and locally full iff it is full and faithful.)
  • locally essentially surjective if for any UU and any zN(U)z\in N(U), there exists a covering family (V iU)(V_i\to U) of UU and x iM(V i)x_i \in M(V_i) such that F(x i)z| V iF(x_i) \cong z|_{V_i}.

Now the “constant sheaf” functor Δ:SetSh 1(C)\Delta \colon Set \to Sh_1(C) induces a morphism of 2-sheaves from the constant 2-sheaf ΔSet\Delta Set to the 2-sheaf Sh 1(C)̲\underline{Sh_1(C)} of 1-sheaves (both of which are large 2-sheaves, but so what). We can see this because there is obviously a functor from the constant 2-presheaf on SetSet to Sh 1(C)̲\underline{Sh_1(C)}, and 2-sheafification is the reflection into the sub-2-category of 2-sheaves. We can thus factor this morphism, using the above ternary factorization system, into three morphisms of 2-sheaves:

ΔSetLocConst 32LocConst 1Sh 1(C)̲. \Delta Set \to LocConst_{\frac{3}{2}} \to LocConst_1 \to \underline{Sh_1(C)}.

It’s fairly obvious that the objects of ΔSet\Delta Set (over some UU) are locally constant sheaves (over UU) in the sense of definition 2. Your point is that the objects of LocConst 1LocConst_1 are the locally constant sheaves in the sense of definition 1. (By construction of the factorization, LocConst 1LocConst_1 is the full sub-2-sheaf of Sh 1(C)̲\underline{Sh_1(C)} determined by the objects which are locally isomorphic to one in the image of ΔSet\Delta Set). The topos being locally connected is more or less equivalent to saying that ΔSetSh 1(C)̲\Delta Set \to \underline{Sh_1(C)} is already full and faithful, so that ΔSetLocConst 1\Delta Set \simeq LocConst_1.

So what is the 2-sheaf I called LocConst 32LocConst_{\frac{3}{2}}? In the familiar classical world, it’s the same as ΔSet\Delta Set—in other words, the morphism ΔSetSh 1(C)̲\Delta Set \to \underline{Sh_1(C)} is already faithful. The question here is about whether the map ΔAut(A)Aut(ΔA)\Delta Aut(A) \to Aut(\Delta A) is monic, which will be true whenever Δ\Delta is sub-cartesian-closed (which, by definition, means that each map Δ(Y X)(ΔY) ΔX\Delta(Y^X) \to (\Delta Y)^{\Delta X} is monic). This is mostly what it means to say that (ΔΓ)(\Delta\dashv \Gamma) is an open geometric morphism, a notion which it is also natural to call “locally (-1)-connected.” Classically, every space, and indeed every topos, is locally (-1)-connected (i.e. “every open set is a union of nonempty ones”), so most of us aren’t used to thinking about the notion very much. But if we were working intuitionistically, or over a base topos other than SetSet, then LocConst 32LocConst_{\frac{3}{2}} might be different from ΔSet\Delta Set and would give us a separate “Definition 32\frac{3}{2}” of locally constant sheaves.

Alternately stated, Definition 1 makes “being locally constant” a property, while classically Definition 2 makes it a structure; but intuitionistically, Definition 2 would make it stuff, leaving room for an intermediate notion where it is only structure.

Similarly, there will be an (n+2)(n+2)-ary factorization system on the (n+1)(n+1)-category of (n+1)(n+1)-sheaves which breaks down the map Δ(n1)GpdSh n(C)̲\Delta (n-1) Gpd \to \underline{Sh_n(C)} into (n+2)(n+2) potentially different notions of “locally constant nn-sheaf.”

I guess you would claim that in all of these cases, the right definition is the final one, which is a full subobject of Sh n(C)̲\underline{Sh_n(C)}. Having written all that, though, I still don’t see any intuitive reason why that’s the right thing to look at. The name “locally constant” certainly suggests that it should be a property, but I don’t think that should bias us unduly. However, I gather that the rest of your comment, and also the paper of Dubuc, present an argument in favor of that viewpoint, so after I digest it perhaps I will have more to say.

Posted by: Mike Shulman on November 28, 2010 4:54 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

I had a closer look at Dubuc’s paper, and, after all, he does not work with all locally contant sheaves. This is due to the fact that, given a covering U1U\to 1 he studies mainly the topos of sheaves which are constant over UU, and this is not well behaved in general. In other words, he focuses on trivialisations of locally constant sheaves over a given covering, but not on locally constant sheaves themselves (he studies sheaves with a `good trivialisation’ over a fixed covering, but, given a locally constant sheaf, he does not study the question of finding a covering over which the sheaf has a `good trivialisation’; this is where genuine Galois theory has its role to play).

Posted by: Denis-Charles Cisinski on November 28, 2010 10:19 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Okay, I think I’m slowly making progress. Part of my confusion was due to the fact that there are two things one might mean by an “action” of a progroup GG on a set AA.

  1. A progroup homomorphism GAut(A)G\to Aut(A), where Aut(A)Aut(A) is regarded as a constant progroup. This is equivalent to regarding GG as a localic group and Aut(A)Aut(A) as a discrete group and asking for a homomorphism of localic groups.

  2. A map G×AAG\times A \to A of pro-sets which is associative and unital. This is equivalent to asking for a continuous map G×AAG\times A\to A of locales which is likewise associative and unital, which in turn is equivalent to asking for a homomorphism GAut(A)G\to Aut(A) of localic groups, where now Aut(A)A AAut(A) \subseteq A^A has the product topology. Note that the product topology on Aut(A)Aut(A) will not be prodiscrete in general.

These are the same when AA is finite, since then the product topology is discrete, but this is not the case in general. For a concrete example where they differ, let GG be the profinite completion of the integers, i.e. the progroup lim/n\underset{\leftarrow}{\lim} \mathbb{Z}/n. A homomorphism from GG to the constant progroup Aut(A)Aut(A) is just an action of /n\mathbb{Z}/n on AA for some nn. However, when Aut(A)Aut(A) has the product topology, then a continuous homomorphism GAut(A)G\to Aut(A) is a \mathbb{Z}-action on AA (i.e. an automorphism of AA) such that every element has finite order.

This starts to look similar to the difference between the two definitions of locally constant sheaves (pointwise eventually constant versus globally eventually constant). Moreover, the intuition connecting Definition 2 of locally constant sheaves to actions of π 1(E)\pi_1(E), as I sketched above, uses the first notion of “action of a progroup(oid),” i.e. a morphism of progroupoids into the (constant pro-)groupoid Core(Set)Core(Set).

However, I can see some arguments for why the second definition of a progroup action is better. For instance, the category of GG-sets with the second definition is itself a Grothendieck topos, which is not the case for the first definition. In the concrete example above, the disjoint union of the orbits n/n\sum_n \mathbb{Z}/n does not admit a GG-action in the first sense, but it does in the second sense. (Unless I am mistaken, though, the category of GG-sets with the first definition is a non-Grothendieck topos, also called the topos of uniformly continuous GG-sets.)

Still thinking…

Posted by: Mike Shulman on November 29, 2010 5:45 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I just had a look at Dubuc’s paper. It looks extremely relevant, thanks! It seems to me like he uses “locally constant” for sheaves satisfying Definition 1 and “covering projection” for those satisfying Definition 2. (His definition of covering projection is not quite the same, but I think it should be equivalent.) He also notes, as we just did, that in general, being a “covering projection” is not a property but a structure.

He defines a topos G UG_U whose objects are “covering projections split by UU,” for any covering family UU in a topos EE, which is “atomic” and equivalent to the topos of actions of a localic groupoid G U\mathbf{G}_U. Then he defines a topos G(E)G(E) as the limit, in ToposTopos, of the topoi G UG_U over refinements of covers, and proves that it is equivalent to the topos of actions of a localic progroupoid, which he calls π 1(E)\pi_1(E). Finally, he proves that this “fundamental localic progroupoid” represents torsors.

Taken at face value, that seems to me to be a vote in favor of Definition 2 being the important one, regardless of which of them we give the name “locally constant” to. In a quick skimming, it doesn’t seem to me as though he makes any use of the “locally constant” sheaves, other than as a category which contains the “covering projections.”

Posted by: Mike Shulman on November 30, 2010 3:29 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

I disagree a little bit with you. I think Dubuc’s paper does not really study what is the nature of locally constant sheaves. What he does looks as follows. He considers a fixed covering U1U\to 1, and he studies locally constant sheaves which have a nice splitting over UU. But we could look at a slightly different question: given a locally constant sheaf XX, it might happen that there exists a covering U1U\to 1 such that XX becomes constant over UU in the sense of definition 1, but not in the sense of definition 2. However, we might hope that there exists another covering V1V\to 1 such that XX becomes trivial over VV in the sense of definition 2. How would you prove that this does not exist? Dubuc’s constructions simply do not address this question at all. In other words Dubuc studies coverings and nice trivialisations over them, but he does not study locally constant sheaves for themselves, so that no general theory of `nice’ locally constant sheaves can be extracted from his work (at least directly). We just don’t care at all if a trivialisation is bad: as we are studying a local property, we just would like that there exists a nice trivialisation!

The idea of my first post here is that there exists another definition of locally constant sheaf: it is a sheaf XX such that there exists a decomposition 1=⨿ iU i1=\amalg_i U_i, and, for each index ii, a group G iG_i as well as a G iG_i-torsor Y iY_i over U iU_i such that XX becomes constant over Y iY_i. This is the definition which is the most related to classical Galois theory (and any locally constant sheaf in this galoisian sense admits a `nice’ splitting). Moreover, with this definition, we must have that LC(Bπ 1(T))LC(T)LC(B \pi_1(T))\simeq LC(T). In other words, for any locally constant sheaf XX in this galoisian sense, there exists a geometric morphism f:TBGf:T\to B G, for GG a (discrete) groupoid, such that XX is in the (essential) image of f *f^*: in other words, this definition is the most related to torsors under a discrete group. I gave a sketch of a proof that any locally constant sheaf in the sense of definition 1 is locally constant in this `galoisian sense’, which means that definition 1 is the relevant one after all, and that it is the optimal statement on the notion of locally constant sheaves.

Posted by: Denis-Charles Cisinski on December 1, 2010 2:14 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Dropping back down a few comment nesting levels…

Definition 1’: a sheaf XX is locally constant if there exists a decomposition 1=⨿ iU i1=\amalg_i U_i as well as epimorphisms V iU iV_i\to U_i, some sets F iF_i, and some isomorphisms V i×XV i×ΔF iV_i\times X\simeq V_i\times \Delta F_i over each V iV_i.

Hmm, that seems like a strange definition to me. Asking for a coproduct decomposition of 1, rather than a cover, doesn’t feel very topos-theoretic. But we can come back to that, because now I am now looking more carefully at your argument for why we can replace coverings by torsors, and I have a different question.

Let YY be the sheaf of isomorphisms from ΔS\Delta S to XX. We know that YY admits a section ss over UU. Define a presheaf Y sY_s on our ambient topos TT as follows. For a sheaf VV on TT, Y s(V)Y_s(V) is the subset of sections tt of Y(V)Y(V) such that there exists a covering f:WU×Vf: W\to U\times V as well as a bijection g:SSg:S\to S, such that the map Y(V)Y(U×V)Y(W)Y(V)\to Y(U\times V) \to Y(W) sends tt to the image of sgs g by the map Y(U)Y(U×V)Y(W)Y(U) \to Y(U\times V) \to Y(W).

I don’t see that Y sY_s is a sheaf unless you replace the bijection g:SSg:S\to S by a section of ΔAut(S)\Delta Aut(S). For instance, if V=V 1V 2V = V_1 \sqcup V_2, then it seems like we could have t 1Y(V 1)t_1 \in Y(V_1) and t 2Y(V 2)t_2 \in Y(V_2) which satisfy that definition using totally different bijections g 1g_1 and g 2g_2, but whose gluing in Y(V)Y(V) will not satisfy it unless we can replace the single gg by a locally constant function into Aut(S)Aut(S).

Also, if f:WU×Vf\colon W \to U\times V is a covering, then Y(U×V)Y(W)Y(U\times V) \to Y(W) must be injective. So saying that tt and sgs g become equal when restricted to WW is the same as saying they become equal when restricted to U×VU\times V, so it doesn’t seem to me that WW is needed. Unless we want to allow gg to be a section only of ΔAut(S)\Delta Aut(S) over WW, rather than over VV or U×VU\times V, which seems reasonable.

However, I still don’t understand why Y sY_s is a torsor. In fact, it seems to me that if Y sY_s is well-supported, then XX satisfies Definition 2. For suppose that VV is well-supported and Y sY_s has a section tt over VV. Then tAut(ΔS)(V)t\in Aut(\Delta S)(V) and we have a cover f:WU×Vf\colon W \to U\times V and a g(ΔAut(S))(W)g\in (\Delta Aut(S))(W) such that f *(π 2 *(t))=f *(π 1 *(s)).gf^\ast(\pi_2^\ast(t)) = f^\ast(\pi_1^\ast(s)) . g.

Now consider W× VWW\times_V W, which covers U×UU\times U. Write s 1s_1 and s 2s_2 for the restrictions of f *(π 1 *(s))Aut(ΔS)(U)f^\ast(\pi_1^\ast(s)) \in Aut(\Delta S)(U) to W× VWW\times_V W along the two projections, and similarly g 1g_1 and g 2g_2 for the restrictions of gg. Crucially, though, t 1=t 2t_1 = t_2, since W× VWW\times_V W has only one map to VV. Then over W× VWW\times_V W we have t=s 1.g 1t = s_1 . g_1 and t=s 2.g 2t = s_2 . g_2, whence s 2 1.s 1=g 2.t 1.t.g 1 1=g 2.g 1 1s_2^{-1} . s_1 = g_2 . t^{-1} . t . g_1^{-1} = g_2 . g_1^{-1}, which lies in ΔAut(S)\Delta Aut(S). However, this is basically saying that XX satisfies Definition 2: its “transition functions” lie in ΔAut(S)\Delta Aut(S) rather than merely Aut(ΔS)Aut(\Delta S).

I got this far last night, but it took me until this morning to figure out what I think is wrong with this argument:

I claim that this implies that Y sY_s is an ΔAut(S)\Delta Aut(S)-torsor. Indeed, this is a local condition, so that, by replacing XX by Y×XY\times X and TT by T/YT/Y, we may assume that X=ΔSX=\Delta S and that ss is the identity of XX. In this case, we have Y s=ΔAut(S)Y_s=\Delta Aut(S), which is the universal ΔAut(S)\Delta Aut(S)-torsor.

I think the issue is that the construction of Y sY_s depends on the choice of UU and s:UYs\colon U\to Y. If in T/YT/Y, you perform the construction starting from U=1 YU=1_Y and the tautological section d:1 YY *Yd\colon 1_Y \to Y^\ast Y, then you will get the universal ΔAut(S)\Delta Aut(S)-torsor. But if you perform the construction starting with s:UYs\colon U\to Y in TT, and then pull back to T/YT/Y, what you’ll get is the same as if you’d performed the construction directly in T/YT/Y but starting with Y *s:Y *UY *YY^\ast s \colon Y^\ast U \to Y^\ast Y instead. And in general, there won’t be any ss such that Y *sY^\ast s can be identified with dd.

Posted by: Mike Shulman on December 5, 2010 5:12 PM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

For the fact that Y sY_s is a sheaf, if I went to fast again, we just have to sheafify it to make it right. The important thing is that Y sY_s defines an Aut(S)Aut(S) torsor. The point is that the inclusion Y sIsom(ΔS,X)Y_s\to Isom(\Delta S, X) is Aut(S)Aut(S)-equivariant. If ss is an isomorphism U×XV×ΔSU\times X\simeq V\times \Delta S over VV, as being an Aut(S)Aut(S)-torsor is a local property, to prove that Y sY_s is a torsor, we may assume that ss is given by a global isomorphism s:ΔSXs:\Delta S\simeq X (by replacing TT by T/VT/V; it seems that I wrote T/YT/Y by mistake in some previous versions). This corresponds to a global section s:1Isom(ΔS,X)s:1\to Isom(\Delta S, X), which gives rise to a canonical Aut(S)Aut(S)-equivariant map ΔAut(S)Isom(ΔS,X)\Delta Aut(S)\to Isom(\Delta S, X). The sheaf Y sY_s is then the image of the latter map. To prove that Y sY_s is an Aut(S)Aut(S)-torsor in TT, it is thus sufficient to prove that the map ΔAut(S)Isom(ΔS,X)\Delta Aut(S)\to Isom(\Delta S, X) is a monomorphism. Working up to isomorphisms, we may assume that X=ΔSX=\Delta S and that ss is the identity; we are thus reduced to prove that the canonical map ΔAut(S)Aut(ΔS)\Delta Aut(S)\to Aut (\Delta S) is a monomorphism.

Posted by: Denis-Charles Cisinski on December 6, 2010 11:17 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

If ss is an isomorphism U×XV×ΔSU\times X\simeq V\times \Delta S over VV, as being an Aut(S)Aut(S)-torsor is a local property, to prove that Y sY_s is a torsor, we may assume that ss is given by a global isomorphism s:ΔSXs:\Delta S\simeq X (by replacing TT by T/VT/V…)

I presume you mean that ss is an isomorphism U×XU×ΔSU\times X\simeq U\times \Delta S over UU, since what you had before was a section of Iso(X,ΔS)Iso(X,\Delta S) over UU. But after that I don’t follow. I thought “being a local property” meant that if we have a cover W1W\to 1 such that W *(Y s)W^\ast(Y_s) is a torsor, then so is Y sY_s. And I believe that the definition of Y sY_s is stable, so that W *(Y s)W^\ast(Y_s) can be identified with Y W *sY_{W^\ast s}.

But if ss is an isomorphism U×XU×ΔSU\times X\simeq U\times \Delta S over UU, then if we pull it back to UU, we get an isomorphism U *U× UU *XU *U× UU *ΔSU^\ast U\,\times_U\, U^\ast X\simeq U^\ast U\,\times_U \,U^\ast \Delta S over U *UU^\ast U in T/UT/U, which is not the same as a global section of Iso(X,ΔS)Iso(X,\Delta S) in T/UT/U; the latter would be an isomorphism U *XU *ΔSU^\ast X \cong U^\ast \Delta S in T/UT/U. Of course there is a global section of Iso(X,ΔS)Iso(X,\Delta S) in T/UT/U, namely ss regarded as a morphism in T/UT/U, but I don’t think that when you regard ss as a morphism in TT and pull it back to T/UT/U, you get the same thing as when you just regard ss as a morphism in T/UT/U. So while you do get (the canonical) ΔAut(S)\Delta Aut(S)-torsor in T/UT/U if you perform the construction starting with ss regarded as a morphism in T/UT/U, I don’t see how to conclude that you also get a ΔAut(S)\Delta Aut(S)-torsor if you perform the construction on U *sU^\ast s, which seems to be what you would need in order to conclude (by “being a local property”) that Y sY_s is also a torsor.

Of course, maybe I’m confused. If you still think I’m confused, can you tell what is wrong with my argument in the preceeding paragraphs deducing Definition 2 from a well-supported Y sY_s?

Posted by: Mike Shulman on December 6, 2010 11:42 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I had time to think about this a little more, and you are right (I couldn’t find a way to really understand this Y sY_s…). And don’t see anyway to save definition 1 any more (even though I really wanted to). For the rest of what I said, it seems that I wanted the definition of locally constant sheaf to be equivalent to the property that there exists a groupoid GG as well as a geometric morphism f:TBGf:T\to BG such that, for any representable object YY in BGBG, XX is constant over f *Yf^* Y. I didn’t want to take this as a definition because, it should be a consequence of the definition in some sense. However, definition 2 certainly (obviously) implies this property (as you said yourself, definition 2 just says that XX is a ΔAut(F)\Delta Aut(F)-torsor). Conversely, any sheaf trivialized by a torsor under a discrete group(oid) defines a covering projection in the sense of Dubuc, so that this goes very much in favour of definition 2. But then, I still have troubles (questions). If we consider LC(T)LC(T) as a full subcategory of TT, then we will have a canonical equivalence LC(Bπ 1(T))LC(T)LC(B\pi_1(T))\simeq LC(T) (I didn’t see any trouble in my proof of this if we consider locally constant sheaves in the sense of sheaves trivialized by torsors under discrete groupoids, unless you have some arguments to destroy it as well!). That is still nice in the sense that this will allow a good characterisation of topoi which are equivalent to the classifying topos of a pro-groupoid, etc. However, this means that locally constant sheaves (considered as a full subcategory of sheaves) do not form a stack on TT in general (unless TT is locally 00-connected). They will form a stack over Bπ 1(T)B\pi_1(T). And this troubles me very much (but maybe I will get used to it).

Posted by: Denis-Charles Cisinski on December 7, 2010 11:55 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

this means that locally constant sheaves (considered as a full subcategory of sheaves) do not form a stack on TT in general… And this troubles me very much.

Well, it seems to me, as I said before and Urs also emphasized, that Definition 2 implies that locally constant sheaves should not, in general, be considered as a full subcategory of sheaves. If we drop that requirement, then they do quite naturally form a stack, namely the constant stack Δ TCore(Set)\Delta_T Core(Set) on the core of SetSet. Moreover, if Core(LC(T))Core(LC(T)) denotes the groupoid Hom T(1,Δ TCore(Set))Hom_T(1,\Delta_T Core(Set)) of locally constant sheaves (the full category of locally constant sheaves would be the sections of the constant 2-sheaf ΔSet\Delta Set rather than the (2,1)-sheaf ΔCore(Set)\Delta Core(Set)), then we tautologically have

Core(LC(T))=Hom T(1,Δ TCore(Set))=Hom Topos(T,BCore(Set))=Hom pro(Gpd)(π 1(T),Core(Set))Core(LC(T)) = Hom_T(1,\Delta_T Core(Set)) = Hom_{Topos}(T, B Core(Set)) = Hom_{pro(Gpd)}(\pi_1(T), Core(Set))

if π 1\pi_1 is defined as a left adjoint to the classifying topos functor BB on pro-groupoids. If BB is moreover fully faithful (is it?), then we would have

Core(LC(T))=...=Hom Topos(Bπ 1(T),BCore(Set))=Hom Bπ 1(T)(1,ΔCore(Set))=Core(LC(Bπ 1(T))). Core(LC(T)) = ... = Hom_{Topos}(B \pi_1(T), B Core(Set)) = Hom_{B \pi_1(T)}(1, \Delta Core(Set)) = Core(LC(B \pi_1(T))).

so that at least on cores, the equivalence between locally constant sheaves on TT and on π 1(T)\pi_1(T) would follow by abstract nonsense. Perhaps one could even boost that up to an equivalence of categories by using 2-topoi instead of (2,1)-topoi.

Posted by: Mike Shulman on December 8, 2010 6:22 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I understand what you wrote perfectly. The reason why I insist on the full subcategory point of view is because I want a recognition theorem: how do you recognize that a topos is the classifying topos of a pro-groupoid? Neither Urs or you did answer this question yet, while it seems to me that the point of view of locally constant sheaves as a full subcategory of sheaves lead to the answer. So it seems to me that the matter of considering locally constant sheaves as a full subcategory of all sheaves depends on what we want to prove. Formulated in another way, according to me, the tautological version of Galois theory is not about an easy adjunction between topoi and pro-groupoids, but about the fact that, given a field kk, the topos of sheaves on the small étale site of kk IS the galois group of kk (if you identify a pro-group(oid) with its corresponding classifying topos); and how do you prove this? Well, you see that finite Galois extensions of kk define a generating family (which just means that any finite separable extension of kk is contained in a Galois extension), and apply a general recognition theorem. This is also related with the fully faithfulness of BB, which seems to hold: if Bπ 1(T)B \pi_1(T) is the category of sheaves on the full subcategory of locally constant sheaves on TT, this implies the fully faithfulness of BB.

For what I understand, it seems (to me) that we need to consider locally constant sheaves both as a full subcategory and as a non full subcategory of all sheaves, depending on what aspect of Galois theory we want to prove and/or to express.

Posted by: Denis-Charles Cisinski on December 10, 2010 2:08 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Okay, I’m sorry, I misunderstood what you were saying. Can you say anything to address my question about why “Galois theory” should be about recognition of classifying topoi of progroupoids?

Also, I don’t think that BB can possibly be fully faithful on all of Pro(Groupoids), since it is not pseudomonic on Pro(Sets). The pro-set +1+1+1\cdots \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N} is not isomorphic to \emptyset (there is, in fact, no morphism of pro-sets from it to \emptyset). But unless I am mistaken, its classifying topos (the limit of Set Set Set \cdots \to Set^{\mathbb{N}} \to Set^{\mathbb{N}} \to Set^{\mathbb{N}}) is trivial, i.e. equivalent to Sh()Sh(\emptyset): a geometric morphism Set \mathcal{E} \to Set^{\mathbb{N}} is a decomposition 1= iU i1 = \coprod_{i\in \mathbb{N}} U_i in \mathcal{E}, and saying that it factors through all the +1+1s implies that U i=0U_i=0 for all ii, hence 1=01=0 and \mathcal{E} is trivial. But maybe if we assume some surjectivity of the transition maps? I asked a related question on MO.

Posted by: Mike Shulman on December 13, 2010 10:52 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

In case anyone is still paying attention to this thread, I think I have found some pro-sets with surjective transition maps for which B:Pro(Set)LocalesB:Pro(Set) \to Locales is not full. I answered my own question at MO. Is this right, do you think? And if so, is non-fully-faithfulness of B a problem for your approach, Denis-Charles? I’m not entirely clear where it entered, or were you were claiming it as a consequence? Or were you saying only that B is fully faithful on pro-groupoids which occur as fundamental progroupoids of toposes? Is there some way to characterize those?

Posted by: Mike Shulman on January 16, 2011 6:17 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

He considers a fixed covering U1U\to 1, and he studies locally constant sheaves which have a nice splitting over UU

That’s what he does in section 2. But then in the following sections he goes on to compare “covering projections” which are split over different coverings, and put them together into a topos that is the action topos of some localic progroupoid.

In other words Dubuc studies coverings and nice trivialisations over them, but he does not study locally constant sheaves for themselves, so that no general theory of ‘nice’ locally constant sheaves can be extracted from his work (at least directly).

I’m really not following. First of all, I am confused about what you mean by sheaves admitting a “nice splitting.” On the one hand you used it to seemingly refer to Dubuc’s “covering projections”, which I believe are the ones satisfying Definition 2. But on the other hand you claimed later that any locally constant sheaf in the sense of Definition 1 is ‘galoisian’ and that these also admit a “nice splitting.” Are these two different kinds of “nice”?

Second, if “nice” means Dubuc’s covering projections, then it seems to me that his work is a general theory of “nice” locally constant sheaves. He doesn’t study “non-nice” locally constant sheaves, but that is only a problem if you have some other reason to believe that those are important, which is the question under discussion. His theory also seems very “Galoisian” to me in that they are classified as certain actions of a fundamental (localic pro)groupoid (but my understanding of the multifarious things people mean by “Galois” is still hazy, so feel free to correct me here). Finally, he also proves that his groupoid classifies torsors for discrete groups, so his theory is also closely related to torsors.

Can you explain further why your theory is better, and how it relates to his? For instance, is your π 1(T)\pi_1(T) the same as his? (It seems like it must be different, since his is a localic progroupoid and you only mentioned ordinary progroupoids… unless the localicness was implicit.)

Posted by: Mike Shulman on December 1, 2010 3:49 AM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

By `nice’, I meant `covering projection’ in the sense of Dubuc. I have a reason to believe (because of the sketched proof) that, for any locally constant sheaf XX in the sense of definition 1, there exists a covering U1U\to 1 such that XX has a nice trivilisation: the reason for this is that this is true for any object of BGB G for any discrete groupoid GG, and that any locally constant sheaf comes from BGB G for some GG, because of the equivalence LC(T)=LC(Bπ 1(T))LC(T)=LC(B \pi_1(T)). The proof uses strongly the fact that any locally constant sheaf in the sense of definition 1 admits a trivialisation in the galoisian sense. This latter property is better because it is a powerful tool to prove things about locally constant sheaves (by reducing to torsors under discrete groups) and because, as I will never insist enough, it is the way to expresss fully and explicitely how the theory of locally constant sheaves is a generalization of classical Galois theory of fields extensions. I will never insist enough: if you don’t speak of trivializations of locally constant sheaves over a covering given by a family of torsors, well you just miss the very meaning of Galois theory.

As for Dubuc’s construction, this is only a partial answer to this story, but this can be easily fixed as follows. He provides a pro-localic groupoid where we would have liked a pro-groupoid. However, by his Theorem 2.19, the classifying topos of this pro-localic groupoid of his is a (filtering) limit of locally connected topoi, so that, by applying the theory of π 1\pi_1 for locally connected topoi, the π 1\pi_1 of Dubuc leads after all a pro-groupoid which represents torsors in our given topos. The definition of the π 1\pi_1 of a topos TT I want is the classifying topos of a pro-groupoid which represents torsors in TT, and it is clear that this is what Dubuc construct after all, with the slight modification I suggested above. So, using Dubuc, you have that π 1(T)\pi_1(T) exists as a pro-groupoid. The proof I sketched above (in my first post) explains why its classifying topos is the category of sheaves on the site given by locally constant sheaves on TT for the canonical topology.

The reason I want the equivalence LC(T)=LC(Bπ 1(T))LC(T)=LC(B \pi_1(T)) is because it provides a proof of the following

Theorem I. Let TT be a Grothendieck topos. Then TT is the classifying topos of a progroupoid if and only if there exists a decomposition 1=⨿ iU i1=\amalg_i U_i in TT, and for each index ii, a group G iG_i as well as a G iG_i-torsor Y iY_i over U iU_i, such that {Y i|iI}\{Y_i \, | \, i\in I\} form a generating family of TT.

What I call a Galois theory is a proof of Theorem I above as well as a proof of the fact that any topos admits a universal map into a classifying topos of a pro-groupoid, and such that we have

Theorem II. A geometric morphism TTT\to T' of Grothendieck topoi iff it induces an equivalence of categories LC(T)LC(T)LC(T')\simeq LC(T).

As it seems that there is no proof of the theorems I and II above for general topoi in the literature, it seems that there is no Galois theory of general topoi in the literature yet in this precise sense.

Posted by: Denis-Charles Cisinski on December 1, 2010 7:33 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

I have a reason to believe (because of the sketched proof) that, for any locally constant sheaf XX in the sense of definition 1, there exists a covering U1U\to 1 such that XX has a nice trivilisation

Does that mean you think definitions 1 and 2 are actually equivalent? I gave what I thought was a counterexample; is it wrong? Or do you disagree that Dubuc’s covering projections are the same as Definition 2?

I will never insist enough: if you don’t speak of trivializations of locally constant sheaves over a covering given by a family of torsors, well you just miss the very meaning of Galois theory.

Unfortunately, just insisting doesn’t convince me — can you additionally explain? (-: When I was an undergraduate, I learned that Galois theory was about a correspondence between subgroups of a Galois group and field extensions. Later I learned to think about sets with an action of the Galois group instead. I’ve gathered that there is an analogy with covering space theory, where covering spaces over a space X are like field extensions and correspond to sets with an action of π 1(X)\pi_1(X). It seems to me that a theorem like Dubuc’s about classifying “covering projections” by actions of a progroupoid π 1\pi_1 is a direct generalization of this.

it is the way to expresss fully and explicitely how the theory of locally constant sheaves is a generalization of classical Galois theory of fields extensions.

Do you mean by that that the classical Galois theory of field extensions can be expressed in terms of locally constant sheaves in some non-locally-connected topos, and that in that case it is Definition 1 which is correct and Definition 2 which is wrong? That’s the only thing I can think of which would make the classical case of field extensions an argument in favor of Definition 1. If so, can you explain? I am not an algebraic geometer and I need these things spelled out very explicitly.

By the way, since Definition 2 is stronger than Definition 1, any sheaf satisfying Definition 2 also satisfies Definition 1, and therefore by your argument also admits a ‘galoisian’ trivialization. So the desirability of this property because it is a powerful tool does nothing to distinguish the two definitions; to choose Definition 1 we need rather an argument that we need nothing besides this property.

Posted by: Mike Shulman on December 1, 2010 4:03 PM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

I am trying to understand your construction of the ‘galoisian’ trivialization.

Let XX be a locally constant sheaf (in the sense of definition 1). Then there exists a (non necessarily unique) decomposition of the terminal sheaf 1= iU i1=\coprod_i U_i and, for each ii a set A iA_i, such that X×U iX\times U_i is locally isomorphic to U i×ΔA iU_i\times\Delta A_i for each ii.

I think I believe that for sheaves on a topological space, but not for sheaves on a locale or topos without enough points—at least, I don’t see how to prove it otherwise. Did you mean to assert it in general?

Here’s how my naive approach would go for topological spaces. By definition 1, we have a covering jW j1\coprod_j W_j \twoheadrightarrow 1 with X×W jX×ΔB jX\times W_j \cong X\times \Delta B_j for each jj and some sets B jB_j. Define jjj\sim j' if B jB jB_j \cong B_{j'}. Clearly this is an equivalence relation; let the ii’s be its equivalence classes, let U iU_i be the support of jiW j\coprod_{j\in i} W_j, and define A i=B jA_i = B_j for any jij\in i (of course it doesn’t matter which). Then X×U iX\times U_i is locally isomorphic to A iA_i (over the cover jiW j\coprod_{j\in i} W_j), and we clearly have iU i1\coprod_i U_i \twoheadrightarrow 1, so it remains to show that the U iU_i are disjoint.

But U iU iU_i \cap U_{i'} is covered by ji,jiW j×W j\coprod_{j\in i, j'\in i'} W_j \times W_{j'}, and X×W j×W jX\times W_j \times W_{j'} is isomorphic to both ΔB j×W j×W j\Delta B_j\times W_j \times W_{j'} and ΔB j×W j×W j\Delta B_{j'}\times W_j \times W_{j'}. Thus, if (the slice topos over) W j×W jW_j\times W_{j'} has a point, then B jB jB_j \cong B_{j'}. Conversely, therefore, if iii\neq i', then B jB jB_j \ncong B_{j'}, so W j×W jW_j\times W_{j'} has no points. If our topos has enough points, this implies that W j×W jW_j\times W_{j'} is initial, and hence so is U iU iU_i\cap U_{i'}.

In general, however, I’m pretty sure it can happen that ΔAΔB\Delta A \cong \Delta B in a nontrivial topos even though ABA\ncong B. For instance, we could consider the classifying topos of bijections between AA and BB. Thus, it seems that in general we could have a covering UV1U \sqcup V \twoheadrightarrow 1 such that X×UΔA×UX\times U \cong \Delta A \times U and X×VΔB×VX\times V \cong \Delta B \times V, but UV0U\cap V \neq 0 and ABA\ncong B, and in such a case I don’t see how to construct your decomposition.

Posted by: Mike Shulman on December 1, 2010 5:19 PM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

Note that I never claimed that definition 1 and 2 are equivalent: I just claim that there is a strong difference, for a given locally constant sheaf XX, between finding a trivialisation in the sense of definition 1 which is not trivialisation in the sense of definition 2, and asserting that there does not exists any trivilasation of XX in the sense of definition 2 at all. So, I just claim that your counter-example is not a reason to reject definition 1.

I will try to explain and jusify my assertions a little more (I really don’t see any real obstruction in any of your arguments yet!).

First, let us consider a locally constant sheaf. Consider a covering ⨿ jW j1\amalg_j W_j \to 1 as well as a trivialisation X×W jΔB j×W jX\times W_j\simeq \Delta B_j\times W_j over W jW_j for each jj. Then define a relation on indices by jjj\sim j' if ΔB jΔB j\Delta B_j\simeq \Delta B_{j'}. This is an equivalence relation. Denote by [j][j] the equivalence class of jj. Define V [j]=⨿ j[j]W jV_{[j]}=\amalg_{j'\in [j]}W_{j'}, and define B [j]=B jB_{[j]}=B_j. I claim that, if ii and jj are not equivalent, than V [i]×V [j]V_{[i]}\times V_{[j]} is empty. To see this, we choose a boolean cover of TT, and, in a boolean topos with axiom of choice, two sets give isomorphic constant sheaves iff they have the same cardinal (as it is a model for classical set theory with ZFC). We then have a trivilisation of shape

X×V [j]ΔB j×V [j]X\times V_{[j]} \simeq \Delta B_j\times V_{[j]}

over V [j]V_{[j]} for each equivalence class [j][j].

In other words, we have proved that, for any locally constant sheaf XX there exists a small indexing set II, a covering of shape ⨿ iU i1\amalg_i U_i\to 1, as well a family of small sets {S i} iI\{S_i\}_{i \in I}, as well as trivialisations of shape

U i×ΔS iU i×XU_i\times \Delta S_i\simeq U_i\times X

over U iU_i for all ii, with the following property: for indices ii and jj, if iji\neq j, then U i×U jU_i\times U_j is empty. Therefore, we must have

1=⨿ iU i,.1=\amalg_i U_i_, .

Now, consider a locally constant sheaf XX, and assume for simplicity that there exists a set SS as well as a covering of shape U1U\to 1 such that U×XΔS×UU\times X\simeq \Delta S\times U over UU. Let YY be the sheaf of isomorphisms from ΔS\Delta S to XX. We know that YY admits a section ss over UU. Define a presheaf Y sY_s on our ambient topos TT as follows. For a sheaf VV on TT, Y s(V)Y_s(V) is the subset of sections tt of Y(V)Y(V) such that there exists a covering f:WU×Vf: W\to U\times V as well as a bijection g:SSg:S\to S, such that the map

Y(V)Y(U×V)Y(W)Y(V)\to Y(U\times V) \to Y(W)

sends tt to the image of sgsg by the map

Y(U)Y(U×V)Y(W).Y(U) \to Y(U\times V) \to Y(W)\, .

In other words, Y sY_s is the subpresheaf of YY which consists of sections which are locally in the orbit of ss under the natural action of Aut(S)Aut(S) on YY. It is clear that Y sY_s is a sheaf (is is defined as a subobject of a sheaf by a local condition). I claim that Aut(S)Aut(S) acts naturally on Y sY_s: if tt is an element of Y s(V)Y_s(V) for some VV, we have to check that, for any element gg of Aut(S)Aut(S), g *(t)=tΔ(g)g^*(t)=t\Delta(g) is in Y s(V)Y_s(V), but this is trivilally true by definition of Y sY_s… Furthermore, as you noticed yourself, any topos is locally 1-1-connected, which implies that the natural map

ΔAut(S)Aut(ΔS)\Delta Aut(S)\to Aut(\Delta S)

is a monomorphism of sheaves. I claim that this implies that Y sY_s is an ΔAut(S)\Delta Aut(S)-torsor. Indeed, this is a local condition, so that, by replacing XX by Y×XY\times X and TT by T/YT/Y, we may assume that X=ΔX=\Delta Sandthat and that sistheidentityof is the identity of X. In this case, we have Y s=ΔAut(S)Y_s=\Delta Aut(S), which is the universal ΔAut(S)\Delta Aut(S)-torsor.

Finally, I claim that XX is trivial over Y sY_s: this is true by definition: Y=Isom(ΔS,X)Y=Isom(\Delta S,X) has a section over Y sY_s.

So we have proved what I claimed from the begining: for any locally constant sheaf XX, there exists a decomposition 1=⨿ iU i1=\amalg_i U_i, a family of sets S iS_i, a family of groups G iG_i, as well as, for each index ii a G iG_i-torsor Y iY_i over U iU_i, such that XX is constant over Y iY_i.

This may be reformulated as follows.

Given a locally constant sheaf XX on a topos TT, there exists a (discrete) groupoid GG and morphism of topoi

f:TBG=G^f:T\to BG=\hat{G}

such that, for any representable presheaf YY over GG, XX is constant in T/f *YT/f^*Y.

I don’t claim that this implies that this implies that any locally constant sheaf has trivialisation in the sense of definition 2 (and in fact, I am realizing that, in order to prove what I claimed, we don’t need it). I just claim that this kind of trivilisation is gentle enough to understand locally constant sheaves. In fact, from there the (sketch of) proof given in my post of November 26th implies that

LC(T)LC(Bπ 1(T))LC(T)\simeq LC(B\pi_1(T))

for any topos TT. The proof of this equivalence uses nothing than the existence of π 1(T)\pi_1(T) (using Dubuc’s paper, if you wish) as well as the fact that we may trivialize locally constant sheaves over torsors as above. This equivalence has the consequence that we have the following result (Theorem I of my post of December 1st was silly):

Theorem. For a topos TT, the following conditions are equivalent. (i) TT is generated by its objects of shape YY, such that there exists a decomposition 1=U⨿V1=U\amalg V as well as a group GG, such that Y×V=Y\times V=\empty and such that YY is a GG-torsor over UU; (ii) TT is generated by a family locally constant sheaves; (iii) TT is the classifying topos of a pro-groupoid.

Proof (assuming the results of my post of November 26th): It is clear that (i) implies (ii) and that (iii) implies (i). Under condition (ii), we must have that LC(T)=LC(Bπ 1(T))LC(T)=LC(B\pi_1(T)) is a site of definition of TT, so that T=Bπ 1(T)T=B\pi_1(T).

This result is part of what I would call a `genuine Galois theory’ because it is a recognition principle for Galois topoi. I mean that for deducing classical Galois theory of fields from its toposic version, you need to say that that the topos of sheaves on the small étale site of a field kk is a Galois topos (i.e. the classifying topos of a pro-groupoid, which, by a connectedness and locally connectedness argument, must be a pro-group), and, then, the abstract results you know about GG-sets translate into classifying results of finite extensions of fields, and thus give you a conceptual proof of Galois’classical results. Such an equivalence of topoi is proved via a recognition principle as above. (Even though the locally connected case is already known and easier to prove.)

Posted by: Denis-Charles Cisinski on December 3, 2010 1:19 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

I just claim that there is a strong difference, for a given locally constant sheaf X, between finding a trivialisation in the sense of definition 1 which is not trivialisation in the sense of definition 2, and asserting that there does not exists any trivilasation of X in the sense of definition 2 at all.

Indeed. That’s what was wrong with the first example that I included in the first version of the post. But my second counterexample was intended to rectify that omission, by giving a sheaf admitting a trivialization satisfying Definition 1, but which fails to satisfy a property that is provably satisfied by any sheaf having any trivialization satisfying Definition 2. (The property is: its restrictions to S 1×{}S^1\times \{\ell\} are eventually constant, up to isomorphism, as \ell increases.)

If this counterexample is correct, then of course all it means is that Definitions 1 and 2 are different; it isn’t by itself a reason to reject either of them.

Posted by: Mike Shulman on December 3, 2010 4:24 AM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

in a boolean topos with axiom of choice, two sets give isomorphic constant sheaves iff they have the same cardinal (as it is a model for classical set theory with ZFC).

I don’t think I believe that either. I believe that in a Boolean topos with choice, one can define (internal) cardinal numbers as special well-orderings, in the usual way, and two objects will be isomorphic iff they have the same (internal) cardinality, and the class of (internal) cardinalities will be well-ordered, and all the other things we are familiar with from ZFC. However, none of that implies that the “constant sheaf” functor from Set to this topos is necessarily “injective on cardinalities.”

In fact, Boolean topoi with choice are essentially the same as what set theorists call “forcing models,” and it’s well-known that in general, forcing can collapse cardinals. The “standard” cardinal-collapsing forcing models are essentially Boolean covers of classifying topoi of the theory of a bijection between two unequal infinite sets, which are the topoi I suggested in my last post as counterexamples.

Posted by: Mike Shulman on December 3, 2010 5:01 AM | Permalink | PGP Sig | Reply to this

Re: Locally Constant Sheaves

I think I should have said, “…Boolean subtoposes of classifying topoi…”. But that doesn’t affect the point.

Posted by: Mike Shulman on December 3, 2010 7:26 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Well, it seems you are right (I am getting used to it). More precisely, my arguments about the fact we may always work with a decomposition of shape 1=⨿ iU i1=\amalg_i U_i are just not correct. However, it is still true that we have the following (which is the abstract version of saying that any polynomial with coefficients in a field kk admits a root in a finite Galois extension of kk):

Theorem. Let XX be a locally constant sheaf over a topos TT (in the sense of definition 1). Then there exists a groupoid GG and a morphism of topoi f:TBGf:T\to BG such that, for any representable presheaf YY in BGBG, XX becomes constant over f *Yf^*Y.

For the proof, one may work first in a relative setting.

Consider a morphism of topoi p:TSp:T\to S as well as a sheaf FF over SS. Say that a sheaf XX over TT is locally isomorphic to FF is there exists a covering U1U\to 1 of TT as well as an isomorphism U×XU×p *FU\times X\simeq U\times p^* F over UU. We may always assume that there exists a sheaf of groups GG over SS as well as a p *(G)p^*(G)-torsor YY in TT, such that Y×XY×p *FY\times X\simeq Y\times p^* F over YY. To prove this, we pick a trivilisation s:U×XU×p *Fs:U\times X\simeq U\times p^* F, we put G=Aut(F)G=Aut(F), and we define YY to be the p *(G)p^*(G)-torsor of isomorphisms from p *(F)p^*(F) to XX which are locally in the orbit of ss.

We may now prove the theorem as follows. Consider a set II, a covering of shape ⨿ iIU i1\amalg_{i\in I} U_i\to 1, a family of sets {F i} iI\{F_i\}_{i\in I}, as well as a family of isomorphisms U i×XU i×ΔF iU_i\times X\simeq U_i\times \Delta F_i over U iU_i. Consider the natural projection

p:(Set/I)× SetTSet/I.p:(Set/I)\times_{Set}T\to Set/I\, .

This means that the pullback of XX in the fiber product (Set/I)× SetT(Set/I)\times_{Set}T is locally isomorphic to F={F i} iIF=\{F_i\}_{i\in I}. Therefore, there exists a group object GG in Set/ISet/I as well as a p *(G)p^*(G)-torsor YY such that XX becomes isomorphic to p *Fp^*F over YY. This implies the theorem above. As I said already, this theorem is the main argument needed to prove that any geometric morphism of topoi TTT\to T' which induces an equivalence of fundamental pro-groupoids induces an equivalence of categories LC(T)LC(T)LC(T')\simeq LC(T).

Posted by: Denis-Charles Cisinski on December 4, 2010 3:04 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Let X be a locally constant sheaf over a topos T (in the sense of definition 1). Then there exists a groupoid G and a morphism of topoi f:T→BG such that, for any representable presheaf Y in BG, X becomes constant over f *Yf^\ast Y.

If that’s true, then it seems to me that we would still get a coproduct decomposition of 1 in T, since the terminal object of BG has a coproduct decomposition corresponding to the component groups of G.

Posted by: Mike Shulman on December 4, 2010 6:31 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Well, I went to fast on the details again. After all, this means that we may have to modify the definition of locally constant sheaves. I propose

Definition 1’: a sheaf XX is locally constant if there exists a decomposition 1=⨿ iU i1=\amalg_i U_i as well as epimorphisms V iU iV_i\to U_i, some sets F iF_i, and some isomorphisms V i×XV i×ΔF iV_i\times X\simeq V_i\times \Delta F_i over each V iV_i.

I already explained the arguments prooving that this definition is equivalent to: there exists a groupoid GG and a morphism of topoi f:TBGf:T\to B G such that, for any representable presheaf YY on GG, XX is constant over f *Yf^*Y.

For the rest, everything I said seems to remain correct.

Posted by: Denis-Charles Cisinski on December 4, 2010 8:38 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

locally constant sheaves should form as stack, and the guys of definition 1 are exactly those which are locally like those of definition 2.

To my mind def 2 is geared towards giving a stack, by definition:

the constant nn-sheaf Δcore((n1)Grpd κ)\Delta core((n-1)Grpd_\kappa) is the nn-sheaf of locally constant κ\kappa-bounded (n1)(n-1)-sheaves.

Def 2 is just unwinding what the sections of the constant nn-sheaf Δcore((n1)Grpd κ)\Delta core((n-1) Grpd_\kappa) are.

I’d think.

Posted by: Urs Schreiber on December 1, 2010 12:37 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

To say what I just said in a more pronounced way:

we have

core(n1)Grpd κ iBAut(A i) core (n-1)Grpd_\kappa \simeq \coprod_{i} \mathbf{B}Aut(A_i)

where the coproduct is over homotopy (n1)(n-1)-types. Since Δ\Delta is left exact, it commutes with looping/delooping and hence also

Δcore(n1)Grpd iBΔAut(A i). \Delta core(n-1)Grpd \simeq \coprod_i \mathbf{B}\Delta Aut(A_i) \,.

Now, every connected object BG\mathbf{B}G in an nn-topos classifies GG-principal nn-bundles.

The canonical nn-permutation representation

ρ:BAut(A i)(n1)Grpd \rho : \mathbf{B} Aut(A_i) \to (n-1)Grpd

induces a notion of ρ\rho-associated nn-bundles.

One can see that def 2 defines a locally constant (n1)(n-1)-sheaf to be, on each connected component, the ρ\rho-associated nn-bundle of an ΔAut(A i)\Delta Aut(A_i)-principal nn-bundles.

Some more discussion of this is here.

Posted by: Urs Schreiber on December 1, 2010 1:08 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I think both definitions give you a stack of locally constant sheaves. The question is whether that should be a full substack of the stack of all sheaves, as I said here.

Posted by: Mike Shulman on December 1, 2010 3:50 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

The question is whether that should be a full substack of the stack of all sheaves,

I was commenting based on my impression – possibly wrong – that Denis-Charles had been suggesting that definition 2 does not give a stack and that stackifying it would yield def 1, when he wrote:

definition 1 will come back anyway: locally constant sheaves should form a stack, and the guys of definition 1 are exactly those which are locally like those of definition 2.

My apologies if my comment was superfluous.

Posted by: Urs Schreiber on December 1, 2010 7:16 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I think he was saying that the full subcategory of sheaves satisfying Definition 2 is not a stack, and that stackifying it gives you Definition 1. That’s what I was trying to get at with the factorization system above: the (leso,ff) factorization system on 2-sheaves is obtained by taking the full image as 2-presheaves, then stackifying.

Posted by: Mike Shulman on December 1, 2010 10:18 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

That’s what I was trying to get at

Okay, I see now. I didn’t follow the exchange closely enough at the beginning.

After dinner today I set myself the exercise of writing a quick but helpful nnLab exposition of how to reformulate the standard Galois theory of field extensions in terms of locally constant sheaves in a suitable topos. I understand the main storyline and was hoping I could just summarize a detailed discussion in the literature. But then I had trouble finding in the literature the actual details that I was hoping for. Apparently there is some text by Gavin Wraith that does what I was looking for, but I didn’t find that. Now I need to do something else.

Posted by: Urs Schreiber on December 1, 2010 11:11 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I hope you get the chance to come back to it and find the details, because I think I would benefit from that a lot.

Fortuitously, however, there is a colloquium here at UCSD tomorrow by Tamas Szamuely entitled “Galois theory: past and present.” So perhaps I will also learn something there.

Posted by: Mike Shulman on December 2, 2010 3:42 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

The most categorical setting for 1-categorical Galois theory is, of course, the general one due Janelidze, a formalism based on an adjunction. See the references at http://ncatlab.org/nlab/show/categorical+Galois+theory

Posted by: Zoran Skoda on December 6, 2010 11:27 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Thanks! I have recently skimmed through the book Galois theories and learned a little bit about that formalism. Perhaps I am missing something, but it seems a little too abstract to give me much understanding of the relationship between Galois theory of fields and of toposes. The hypotheses and definitions seem to be tailored “just so” as to make the theorem true in the most generality. What I would really like is a careful, clear explanation of how Galois theory of fields can be extracted from Galois theory of toposes as a special case (which I’ve gathered from various comments must be the case), rather than how to view both as special cases of something more general.

(Szamuely’s colloquium was good, but mostly historical about what Galois actually wrote and how it gradually changed into what we know today as Galois theory (of fields). He mentioned fundamental groups and covering spaces at the end, but only as an analogy, and didn’t use the word “topos.”)

This comment contains the best explanation I’ve heard so far:

for deducing classical Galois theory of fields from its toposic version, you need to say that that the topos of sheaves on the small étale site of a field k is a Galois topos (i.e. the classifying topos of a pro-groupoid, which, by a connectedness and locally connectedness argument, must be a pro-group), and, then, the abstract results you know about G-sets translate into classifying results of finite extensions of fields, and thus give you a conceptual proof of Galois’ classical results.

It sounds like the claims are that (1) locally constant sheaves on the small étale site of a field k have something to do with field extensions of k, and (2) the fundamental (pro)group of that (connected and locally connected?) topos has something to do with the (absolute?) Galois group of k. But I would still like more details, or a pointer to where I can read about them. It seems incredible to me that an explanation of this is so hard to find in the literature, given how frequently people bandy about the adjective “Galois” when discussing fundamental groups of toposes. But perhaps we are just not looking in the right places.

Posted by: Mike Shulman on December 6, 2010 3:37 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I bet Minhyong Kim could help. Does he have anything online? There was a very nice explanation of Galois groups and fundamental groups here. I wonder if there’s anything explicitly topos theoretic. Got to dash, but did just see a topos crop up here.

Posted by: David Corfield on December 6, 2010 3:54 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Thanks! I don’t see any topoi in the MO answer. It seems to be a very different point of view to me, because it is all about the importance of the basepoint, whereas from the topos point of view (at least, the way it seems to me) choosing a basepoint is unnatural—the canonically defined object is the fundamental groupoid.

Another answer recommended this book, though, which I have not picked up yet. Let me see if I can get my hands on it.

Posted by: Mike Shulman on December 6, 2010 5:33 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I was asking Minhyong about the idea of treating all points in a fundamental groupoid here.

Posted by: David Corfield on December 6, 2010 6:23 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Let’s see, I think I remember reading that the etale topos of a field kk is the classifying topos for separable closures of kk. So in particular, its “points” in the topos-theoretic sense are exactly the things that Minhyong is saying should be used as “basepoints” when computing fundamental groups. However, the etale topos of a more general scheme should classify something more general than separable closures of its function field. Separable closures of residue fields at arbitrary points (of the underlying topological space)? Are the “geometric points” a subclass of these?

I think all of these toposes have “enough points,” so that we have a surjective geometric morphism Set PSh(X et)Set^P \to Sh(X_{et}) for PP the set of points. Since Sh(X et)Set π 1(X)Sh(X_{et}) \to Set^{\pi_1(X)} is also supposed to be surjective (in fact, connected), we should have a surjection Set PSet π 1(X)Set^P \to Set^{\pi_1(X)}, which should allow us to regard π 1(X)\pi_1(X) as a (pro)groupoid with PP as its set of objects. But if Sh(X et)Sh(X_{et}) is itself connected, then the groupoid π 1(X)\pi_1(X) is also supposed to be connected, so that if we want to compute a fundamental group, it shouldn’t matter which of these points we pick (up to an isomorphism that is unique up to conjugation). In particular, that seems to imply that the fundamental group at a non-generic or non-geometric point should be the same as at a generic or geometric one (for a scheme whose etale topos is connected).

But I’m on pretty shaky ground here.

Posted by: Mike Shulman on December 6, 2010 11:19 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

It sounds like the claims are that (1) locally constant sheaves on the small étale site of a field k have something to do with field extensions of k, and (2) the fundamental (pro)group of that (connected and locally connected?) topos has something to do with the (absolute?) Galois group of k.

Here is something, taken from page 11 of

  • H. W. Lenstra, Galois theory for schemes , Mathematical Institute of the University of Amsterdam (1985)

First the statement about the relation between étale coverings of schemes and their fundamental group. This is not formulated topos-theoretically in these lectures, but I think we know how to do that: the locally constant sheaves in the petit topos of the scheme should give the étale covers.

So we have

Theorem Let XX be a connected scheme. Then there exists a profinite group π 1(X)\pi_1(X), uniquely determined up to isomorphism, such that the category FEt XFEt_X for finite étale coverings of XX is equivalent to the category π 1(X)Set\pi_1(X) Set of finite sets on which π 1(X)\pi_1(X) acts continuously.

And then the examples relating this to classical Galois theory of field extensions are the following.

Example

If KK is an arbitrary field, then π 1(SpecK)\pi_1(Spec K) is the Galois group of the separable closure of KK over KK.

Let X=SpecAX = Spec A, where AA is the ring of integers in an algebraic number field KK. Then π 1(X)\pi_1(X) is the Galois group of MM over KK, where MM is the maximal algebraic extension of KK that is unramified at all non-zero prime ideals of AA.

More generally, if aAa \in A, a0a \neq 0, then π 1(SpecA[1a])\pi_1(Spec A[\frac{1}{a}]) is the Galois group, over KK, of the maximal algebraic extension of KK that is unramified at all non-zero prime ideals of AA not dividing aa.

Posted by: Urs Schreiber on December 6, 2010 3:59 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

the locally constant sheaves in the petit topos of the scheme should give the étale covers.

Can you say anything about how that happens? I think this is the crux of what I need to understand in order to get to point (1). I’d settle for understanding the case of fields, since I don’t really understand étale maps of general schemes, but a finite étale cover of Spec of a field kk is, I read, just a finite disjoint union of Specs of finite separable extensions of kk.

I guess that given any étale cover YXY\to X, I can take its sheaf of sections. Is that sheaf locally constant? And why does any (presumably finite) locally constant sheaf arise in that way?

Given this identification, then the theory of fundamental group(oid)s of toposes gives us a profinite group whose finite actions are equivalent to finite locally constant sheaves, hence to finite étale morphisms, which over a field are finite disjoint unions of finite separable extensions. Evidently the “finite disjoint unions” correspond to the decomposition of any finite G-set as a finite disjoint union of transitive ones.

So what remains for point (2) is to identify this progroup with the usual definition of the Galois group, i.e. (in the field case) the automorphisms of the separable closure over kk. I’m going to speculate that the separable closure determines another étale sheaf on Spec(k)Spec(k), which is not finite (though maybe it is locally constant?), and that this sheaf plays the role of a “universal cover” in that any (finite?) locally constant object is trivialized over it. Assuming the étale topos of kk is locally connected, that should imply that the (profinite) fundamental groupoid can be identified with (the profinite completion of) the image under Π 0:Sh(k et)Set\Pi_0 \colon Sh(k_{et}) \to Set of the Cech nerve of this “universal cover.” How do we get from there to automorphisms of the separable closure?

Posted by: Mike Shulman on December 6, 2010 6:04 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

the locally constant sheaves in the petit topos of the scheme should give the étale covers.

Can you say anything about how that happens

I’ll try to give a good answer a little later. For the moment I’ll just record the statement of this fact at nnLab: étale morphism – As locally constant sheaves.

Posted by: Urs Schreiber on December 6, 2010 7:14 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Hmm, that last comment was meant to show up down here, but I guess I clicked “Reply” on the wrong comment.

While I’m doing my best to poke holes in everything you say, let me also say thanks a lot for working through this with me (and for being willing to keep posting despite my trying to poke holes in everything—other lurkers take note!). I’m learning a bunch, and it will be great to get all of this settled correctly (and then we can nLabify it and move on to the ∞-case).

Overall, I gather that your “design criterion” for a definition of locally constant sheaves (or one of them) is that it should give a good characterization of Galois toposes—whatever those are. Is there an accepted general definition of “Galois topos” that applies in the non-locally-connected case as well, or should that definition be regarded as up for debate also? Should a Galois topos be defined as one which is the classifying topos of a progroupoid? Or as one which is the classifying topos of its fundamental progroupoid (is that different)? Or as one which is generated by locally constant objects (in which case, the definition might depend on the notion of “locally constant” one adopts)?

Posted by: Mike Shulman on December 6, 2010 6:04 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Dubuc has another paper, “On the representation theory of Galois and Atomic Topoi”, which seems to me to be something of a prerequisite to his “Fundamental Pro-Groupoid” paper. Sections 5 and 3 seem very relevant to this discussion, dealing with locally connected and galois objects, and just what types of topoi are characterized as classifying topoi of pro-discreet groupoids.

Posted by: tjs on December 7, 2010 3:06 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

Thanks! I am currently very confused about the relationship between localic things and pro- things in this situation. From the higher-categorical and shape-theory pictures, pro-groupoids seem to be the most natural thing to consider, and the fact that they can (sometimes) be represented by taking honest limits of locales seems a convenient accident. But Dubuc seems to be solidly in the camp of believing that the localic things are fundamental, and occasionally using localic things that are not prodiscrete. Anyone have anything to say to alleviate my confusion?

Posted by: Mike Shulman on December 8, 2010 5:56 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Well, I think the issue is not pro vs. localic, but pro-groupoid vs. pro-localic groupoid.

If you take at what Dubuc does in a pointed situation, you would get, in general, a pro-localic group as a fundamental group, because connected atomic topoi are the classifying topoi of localic groups. And so in general you would get a pro-localic groupoid whose space of objects is pro-discrete, or something like that. And this is what Dubuc gets by trying to generalize the galoisian picture of the fundamental group to the general topos situation.

Then there is the shape theory picture of representing a hom-funtor by a pro-groupoid. So what is the relationship between these two constructions? A start on this question, I think, would be to ask what would be the fundamental pro-groupoid (in the shape theory sense) of a simple non-galois atomic topos like the classifying topos of U(1)? Actually, this topos is locally connected, so it should had an honest fundamental groupoid according to the shape theory picture, right? Anyone know what it would be?

Posted by: tjs on December 8, 2010 6:30 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

In what sense do you mean the “classifying topos” of U(1)U(1)? Since U(1)U(1) is connected, it cannot act nontrivially on a discrete set, so the category of continuous U(1)U(1)-sets is just SetSet, whose fundamental groupoid is trivial. It probably has a nontrivial classifying (,1)(\infty,1)-topos, but it doesn’t sound like that’s what you meant.

Posted by: Mike Shulman on December 8, 2010 6:46 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Oops! I was thinking “What’s the easiest non-discrete topological group” but did not think “which has a non-trivial topos of equivariant sheaves.” What’s an easy example of a non-galois atomic topos to test these ideas on? I really don’t have a clue about atomic topoi, but Dubuc’s work has made me think they are important to this whole Galois theory/homotopy theory story. Then again maybe not…

Posted by: tjs on December 8, 2010 7:29 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

It seems to me that the only reason Dubuc ends up with something pro-localic instead of something merely pro- is that he only half-trivializes his locally constant (Def. 2) sheaves. That is, he considers a covering family U=(U i)U=(U_i) and the category 𝒫 U\mathcal{P}_U of “locally constant sheaves (Def. 1) trivialized by UU”, with a subcategory 𝒢 U\mathcal{G}_U of the “covering projections (Def. 2)”. Then he identifies 𝒢 U\mathcal{G}_U with the classifying topos of a localic groupoid, and then takes the limit over refinements of UU to get a pro-localic groupoid.

However, I would be inclined to further break down the categorgies 𝒢 U\mathcal{G}_U, by replacing the covering family (U i)(U_i) by a 1-bounded hypercover, i.e. in addition to (U i)(U_i) we include as data a covering (V ijk)(V_{i j k}) of each U iU jU_i \cap U_j. Then a locally constant object (Def. 2) is “trivialized over (U,V)(U,V)” if it becomes constant over each U iU_i, and the transition functions from U iU_i to U jU_j become constant over each V ijkV_{i j k}. (Rather than merely, as in Dubuc’s definition, asking for V ijkV_{i j k} to exist, we give it as extra data.) It seems to me that the category of such things should be equivalent to the classifying topos of the fundamental groupoid of the nerve of the hypercover, which is an ordinary groupoid. Then we should only need to take one limit, over hypercovers, to get a plain pro-groupoid.

Posted by: Mike Shulman on December 8, 2010 6:59 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I have been thinking about how to define the fundamental 2-group(oid) of a scheme using a category of locally constant stacks on the etale site. In SGA1 (or Lenstra’s notes) there are axioms for “Galois categories / toposes” that recognise when a category is equivalent to a category of continuous representations of a profinite group. Does anyone here know of a similar set of axioms for recognising 2-categories which are equivalent to 2-categories of continuous representations of a profinite 2-group? In other words, does anyone know the definition of a Galois 2-topos?

Posted by: JHH on December 8, 2010 12:35 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Well, even if you had the complete picture of fundamental 2-groupoids of 2-topoi, there is still the issue of using the right topos for the job. For example, if you use the Zariski topos of a scheme you get the right connected components but not the right fundamental group. So why should the etale topos give you the right fundamental 2-groupoid?

You know you are doing it “right” by comparing it to the complex case. First, there is a topological notion of component, and the Zariski components and the components in the complex topology agree. Second, there is the topological theory of covering spaces, which give a theory of the fundamental group (with no mention of topoi…yet). Etale maps of complex varieties give the same category of as finte topological coverings, so you can get a profinite fundamental group out of this. Now, covering spaces are equivalent to locally constant sheaves, and etale maps are not analogous to any kind of Zariski sheaf. So if you want a good category of sheaves to get a good cohomology theory (like Grothendieck did), you need to make etale maps into sheaves. Hence the etale site and etale topos.

OK, that much of the story is common knowledge. But what about going up to the next level? First you would need to know what a 2-covering space is in the topological sense. It would have to be a map of topological stacks/topological groupoids. Then you would need to find an algebraic analogue of such a stack map that agrees with the topological notion in the case of complex varieties, subject to some finiteness issues (but first you would need to make sure you were using the “right” notion of algebraic stack…). Hopefully, these would form, or at least generate, some type of “Galois-2-Category”, and then using the mythical yoga of galois-2-categories you would get the “correct” algebraic fundamental 2-groupoid.

Posted by: tjs on December 8, 2010 7:04 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

First you would need to know what a 2-covering space is in the topological sense. It would have to be a map of topological stacks/topological groupoids.

I believe I can be of some assistance here :). My PhD thesis was on precisely this thing. I worked with topological groupoids and internal anafunctors instead of topological stacks.

There is a way to get from 2-covering spaces to locally constant stacks, and I believe there is a way back, but I haven’t given it serious thought (yet). I’ll be happy to discuss! (my email can be found by finding me on the nLab if you want it)

Posted by: David Roberts on December 8, 2010 11:58 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

In other words, does anyone know the definition of a Galois 2-topos?

Just a comment: the usual definition of Galois topos assumes local connectivity. If we allow ourselves that, then for Galois (2,1)(2,1)-toposes we might assume local 11-connectivity. And for a Galois (,1)(\infty,1)-topos local \infty-connectivity. If we have that, then there is a good topos theoretic definition of the full fundamental \infty-groupoid, described at geometric homotopy groups in an \infty-topos.

The discussion here is all motivated by the question how to generalize this nice story for the case when things are not (or not highly) locally connected. So I guess to give a good definition of Galois (2,1)(2,1)-topos one should first see what the good definition of a Galois 1-topos in the not-locally connected case is. I think this is what Mike has been driving at here.

For what it’s worth, for the geometric applications that myself am looking at, I am not interested in handling non-locally \infty-connected (,1)(\infty,1)-toposes, but conversely in finding as many good big locally \infty-connected (,1)(\infty,1)-toposes as possible. A class of examples is provided by \infty-stacks over (,1)(\infty,1)-cohesive sites. The examples that I know of include topology and (synthetic) differential geometry. I am still hoping to find their cohesive derived versions and a version for algebraic geometry. If such existed, the above definition for geometric homotopy groups in an \infty-topos would also apply to schemes / derived schemes.

Posted by: Urs Schreiber on December 8, 2010 2:06 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

then there is a good topos theoretic definition of the full fundamental ∞-groupoid, described at geometric homotopy groups in an ∞-topos.

When discussing the fundamental ∞-groupoid of a topos, I would prefer to point people to fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos, which is specifically about that.

For locally 1-connected (2,1)-topoi, that general picture should give the right notion of fundamental groupoid. In particular, that fundamental groupoid will classify locally constant (2,1)-sheaves (= stacks of groupoids), essentially by definition.

I guess to give a good definition of Galois (2,1)-topos one should first see what the good definition of a Galois 1-topos in the not-locally connected case is. I think this is what Mike has been driving at here.

Yes, that’s about right. Note that the existing large amount of interest in 1-topoi that are not locally 1-connected, hence whose fundamental groupoids are pro- or localic groupoids, implies there should be an equal amount of interest in 2-topoi that are not locally 1-connected (since 2-sheaves on a non-locally-1-connected 1-topos will be a non-locally-1-connected 2-topos). So even if one finds non-locally-connected 1-topoi uninteresting, working out Galois theory for them is useful because as we go up in dimensions, the local connectivity hypotheses would become more and more restrictive.

(Is the etale site locally 1-connected?)

In particular, I expect that, as Urs suggests, the classical characterization of locally-connected Galois 1-toposes could be categorified to give a characterization of locally-1-connected Galois (2,1)-toposes. Although I, personally, am far from understanding enough about the classical notion to write down any such categorification myself.

Posted by: Mike Shulman on December 8, 2010 5:52 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

When discussing the fundamental ∞-groupoid of a topos, I would prefer to point people to fundamental \infty-groupoid of a locally \infty-connected (,1)(\infty,1)-topos, which is specifically about that.

Yes, that’s because you think small and I think big ! ;-)

Seriously: I agree of course. But when JHH metioned the fundamental groupoid of a scheme, I was internally thinking of looking at that scheme as a 0-truncated \infty-stack over some version of the big étale site, hence as an object in a big \infty-topos of algebraic geometry. Whereas you are immediately thinking of looking at its small \infty-topos.

But of course the big perspective here hinges on having a sufficiently highly locally connected big topos. One needs to ask, as you do

(Is the étale site locally 1-connected?)

or if there is some variant of it that is. I am too uneducated to know the answer, still, but maybe I find time to think about it. (But currently I am more absorbed with thinking about whether the derived Cahier topos is \infty-cohesive :-)

Posted by: Urs Schreiber on December 8, 2010 7:21 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

I am in the process of adding at nnLab: Galois theory details on classical Galois theory and its geometric re-interpretation, by taking material from Hendrik Lenstra’s 1985 lecture in Amsterdam.

Posted by: Urs Schreiber on December 6, 2010 6:36 PM | Permalink | Reply to this

Re: Locally Constant Sheaves

Just for the record, probably no news here:

while looking at Galois theory for the locally simply connected case I noticed the article

Barr, Diaconescu, On locally simply connected toposes and their fundamental groups (NUMDAM)

This is to some extent orthogonal to the article by Dubuc that was already discussed above, since here particularly nice assumptions on the topos are made. They also discuss two examples of fundamental groups of “pathological” topological spaces.

In any case, I included the reference at the entry-under-construction locally constant sheaf.

Posted by: Urs Schreiber on December 7, 2010 12:50 AM | Permalink | Reply to this

Re: Locally Constant Sheaves

I have a general question about “Galois theory.” Many people, including Denis-Charles here and Dubuc’s papers that have been cited, seem to say that Galois theory is about characterizing “Galois toposes,” which are the classifying toposes of some kind of (profinite, prodiscrete localic, or pro-) group (oids). But I still don’t understand why such a characterization is important to recover the traditional meaning of “Galois theory.”

My current understanding is that we recover traditional Galois theory by working with the etale site of a field kk, in which (1) locally constant sheaves can be identified with some field extensions, and (2) the fundamental (pro-)group of that topos is the absolute Galois group of kk. It seems to me that once we have these facts, the correspondence between field extensions and Gal(k)Gal(k)-actions requires only knowing that locally constant sheaves in a topos EE are classified by functors π 1(E)Set\pi_1(E)\to Set. Moreover, this follows tautologically from Definition 2 of locally constant sheaves and a definition of π 1\pi_1 as left adjoint to the “classifying topos” functor from (pro) groupoids, along with the fact that global sections of a constant stack on a groupoid GG are equivalent to geometric morphisms into its classifying topos.

So why is it important, for purposes of “recovering classical Galois theory,” to know that the whole etale topos of a field Sh(k et)Sh(k_{et}) is equivalent to the classifying topos of π 1(Sh(k et))\pi_1(Sh(k_{et})), rather than just saying something about the locally constant objects?

Posted by: Mike Shulman on December 8, 2010 6:04 PM | Permalink | Reply to this

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