Locally Constant Sheaves

November 25, 2010

Posted by Mike Shulman

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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?

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

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

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

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

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 $G$ is a groupoid, then a global section of the constant stack $\Delta G$ is determined by

an open cover $(U_i)$ of $X$ ,
for each $i$ , an object $a_i\in G$ ,
for each $i,j$ , a locally constant function $g_{i j}\colon U_i \cap U_j \to Hom_G(a_i,a_j)$ ,
such that $g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x)$ for any $i,j,k$ and any $x\in U_i \cap U_j\cap U_k$ .

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

an open cover $(U_i)$ of $X$ ,
for each $i$ , a set $A_i$ ,
for each $i,j$ , a locally constant function $g_{i j}\colon U_i \cap U_j \to Iso(A_i,A_j)$ ,
such that $g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x)$ for any $i,j,k$ and any $x\in U_i \cap U_j\cap U_k$ .

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

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

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 $\Delta$ preserves cartesian products, and $Set$ and $Sh(X)$ are both cartesian closed, for any sets $A$ and $B$ there is a canonical map $\Delta(B^A) \to (\Delta B)^{\Delta A}$ . If this map is always an isomorphism, we call $\Delta$ a cartesian closed functor. And the “object of isomorphisms” $Iso(A,B)$ in any cartesian closed category can be constructed as an equalizer of a pair of maps $A^B \times B^A \rightrightarrows A^A \times B^B$ , and since $\Delta$ also preserves equalizers, if it is a cartesian closed functor, then we have $\Delta Iso(A,B) \cong Iso(\Delta A,\Delta B)$ for any sets $A$ and $B$ .

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

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

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

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

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

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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 $X$ is a functor $\Pi_1(X) \to Core(Set)$ .

For general $X$ , and with the “usual” definition of $\Pi_1(X)$ in terms of paths, I don’t think this is equivalent to either of the first two definitions. For instance, if $X$ is something like the Warsaw circle, then $\Pi_1(X)$ is trivial when defined in terms of paths, but there are nonconstant locally constant sheaves on $X$ 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 $X$ is locally 1-connected, then it has a fundamental groupoid $\Pi_1(X)$ 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: $X$ being locally 1-connected means that the constant stack functor $\Delta\colon Gpd \to Sh_{(2,1)}(X)$ has a left adjoint $\Pi$ , so that global sections of $\Delta Core(Set)$ are equivalent to groupoid maps $\Pi(\ast)\to Core(Set)$ . Thus defining $\Pi_1(X)\coloneqq \Pi(\ast)$ works.

If $X$ 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 $Gpd \to Gpd$ 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 $X$ 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

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

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

But I neglected to add the requirement that the corresponding function $f^\infty$ (the endomorphism of the stalk over the point at infinity), which is defined by $f^\infty(x) = f_x(n)$ for sufficiently large $n$ , 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 $A=\mathbb{N}$ and $f_x(n) = \begin{cases} x+1 &\quad if x\lt n\\ 0 &\quad if x=n\\ x &\quad if x\gt n \end{cases}$ Then each $x\mapsto f_x(n)$ is an automorphism, and each sequence $(f_x(n))_{n\in \mathbb{N}}$ is eventually constant at $x+1$ , so $f^\infty(x) = x+1$ , which is not an automorphism.

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

Secondly, I also wrote:

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

However, I claim that $U\cap V$ is homeomorphic to an open interval times the one-point compactification of $\mathbb{N}$ . (Draw a picture.) Thus, if $A$ is an infinite set, we can pick an automorphism of $\Delta_{U\cap V} A$ which is constant in the open interval direction, and does not come from a global section of $\Delta_{U\cap V} Aut(A)$ . The sheaf on $X$ resulting from gluing $\Delta_U A$ to $\Delta_V A$ 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 $F$ and $G$ are obtained by gluing $\Delta_U A$ to $\Delta_V A$ along some pointwise-eventually-constant sequence of automorphisms $f_n\in Aut(A)$ and $g_n\in Aut(A)$ , as above, then an isomorphism $F\cong G$ would be given by automorphisms of $\Delta_U A$ and $\Delta_V A$ which commute with the $f$ ’s and $g$ ’s.

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

I think this means that any sheaf $F$ on $X$ satisfying Definition 1 is actually isomorphic to the constant sheaf $\Delta_X A$ . Of course taking $G=\Delta_X A$ means that $g_n = \id$ for all $n$ . Now I claim that for any $F$ , we can define the $h$ ’s and $k$ ’s inductively as follows: let $k_0=\id$ , then we must have $h_0 = f_0^{-1}$ , then $k_1 = h_0 f_1$ , then $h_1 = k_1 f_2^{-1}$ , then $k_2 = h_1 f_{3}$ , and so on. These $h$ ’s and $k$ ’s and their inverses are pointwise eventually constant because we have $h_{n+1} = h_n f_{2n+1} f_{2n+2}^{-1}$ and so on, and the $f$ ’s and their inverses are pointwise eventually constant.

In retrospect, this is not surprising, since the space $X$ intuitively “has no loops,” so from the $\Pi_1$ 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

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How about looking at $p$ -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

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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 $S^1$ is connected and locally 1-connected with $\Pi_1(S^1)=\mathbb{Z}$ , all the definitions agree over $S^1$ , and a locally constant sheaf over $S^1$ is determined by a set $A$ equipped with an automorphism $f$ . Given such, we can construct the locally constant sheaf explicitly by writing $S^1=U\cup V$ , where $U$ and $V$ are homeomorphic to open intervals and $U\cap V$ is homeomorphic to a disjoint union of two open intervals. Then we glue together $\Delta_U A$ and $\Delta_V A$ along $1_A$ and $f$ on the two components of $U\cap V$ , respectively.

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

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

To see this, suppose $G$ satisfies Definition 2. Then, in particular, $X$ is covered by open sets over which $G$ is isomorphic to constant sheaves. If $G$ is going to be isomorphic to $F$ , then these constant sheaves will all have to be constant at the set $A$ .

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

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

Moreover, since all the gluing data is constant after $k$ , for any $\ell\gt k$ the restriction of $G$ to $S^1\times \{\ell\}$ must be isomorphic to $H$ . However, the sheaf $F$ does not satisfy this property for any locally constant sheaf $H$ on $S^1$ , since the restrictions of $F$ to $S^1\times \{\ell\}$ are determined by the automorphisms $(f_n)$ . Since these are not eventually constant, they determine a sequence of locally constant sheaves on $S^1$ 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

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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

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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 $X$ be a locally constant sheaf (in the sense of definition 1). Then there exists a (non necessarily unique) decomposition of the terminal sheaf $1=\amalg_i U_i$ , and, for each $i$ a set $A_i$ , such that $X\times U_i$ is locally isomorphic to $U_i\times \Delta A_i$ for each $i$ . By working over each $U_i$ , we see that, for the study of $X$ , we may assume without loss of generality that there exists a set $A$ such that $X$ is locally isomorphic to $\Delta A$ . Pick a covering $V\to 1$ , and an isomorphism $s:V\times\Delta A\to V\times X$ over $V$ . Denote by $Y$ the sheaf of isomorphisms from $\Delta A$ to $X$ , and write $G=Aut(A)$ for the set of bijections from $A$ to itself. Write $Y_s$ for the subsheaf of $Y$ whose sections are those which are locally in the orbit of our given trivialization $s\in Y(V)$ under the natural action of $G$ (i.e. a section $t\in Y(U)$ belongs to $Y_s(U)$ if there exists an epimorphism $p:W\to U\times V$ and an element $g$ of $G$ , such that $t.g$ and $s$ have the same image in $Y(W)$ ). Then $Y_s$ is a $\Delta G$ -torsor, and $Y_s\times X$ is constant over $Y_s$ . In other words, we have the following nice property:

If $X$ is locally constant sheaf, there exists a decomposition of the terminal sheaf $1=\amalg_i U_i$ such that, for any index $i$ , there exists a group $G_i$ as well as a $G_i$ -torsor $Y_i$ over $U_i$ with the property that $X\times Y_i$ is constant over $Y_i$ .

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

$\pi_1 : Topoi \rightleftarrows Pro(Groupoids) : B$

where, for a pro-groupoid $G=lim_i G_i$ , $BG$ is the $2$ -limit of the topoi $BG_i=\hat{G}_i$ , and where $\pi_1(T)$ is the pro-groupoid associated to $T$ (whatever it might be).

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

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

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

$\pi_1(T)\to \pi_1(T')$

is an equivalence of topoi.

We then have a kind of toposic version of Quillen’s theorem A (with respect to $1$ -equivalences): consider a morphism of topoi $f:T\to T'$ such that there exists a generating family $G$ of $T'$ satisfying the following assumption: for any $X$ in $G$ , the morphism $T/f^*X\to T'/X$ is a $1$ -equivalence. Then $f$ itself is a $1$ -equivalence. (The proof is essentially the same as for classical theorem A: one shows that $T$ is the homotopy colimit of the $T/f^*X$ ’s).

This implies the following

Proposition: Assume that $f:T\to T'$ is a $1$ -equivalence. Then, for any locally constant sheaf $X$ on $T'$ , the map $T/f^*X\to T'/X$ is a $1$ -equivalence as well.

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

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

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

Corollary: If $f:T\to T'$ is a $1$ -equivalence, then the inverse image functor $f^*:LC(T')\to LC(T)$ is an equivalence of categories.

Let us check that $f^*$ is fully faithful on locally constant sheaves. Let $X$ and $Y$ be locally constant sheaves over $T'$ . The property that the map

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

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

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

$LC(B\pi_1(T))\simeq LC(T)\, .$

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

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

Re: Locally Constant Sheaves

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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 $f:T\to T'$ and $q:T'\to S$ be geometric morphisms of topoi, and write $p=q f$ . Assume that there is a generating family $G$ of $S$ such that, for any $X$ in $G$ , the map

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

is a $1$ -equivalence, then the map $f$ is a $1$ -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 $S=B\pi_1(T')$ .

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

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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 $Cat$ , except that everything is “localized” so we get (locally faithful, locally full, locally essentially surjective). A map $F\colon M\to N$ of 2-sheaves is:

locally faithful if for any $U$ , any $x,y\in M(U)$ and $\alpha,\beta \in Hom_{M(U)}(x,y)$ , if $F(\alpha)=F(\beta)$ , then there is a covering family $(V_i\to U)$ of $U$ such that $\alpha|_{V_i} = \beta|_{V_i}$ for all $i$ . (Since $M$ is a 2-sheaf, this is equivalent to $\alpha=\beta$ , so localizing faithfulness doesn’t change it.)
locally full if for any $U$ , any $x,y\in M(U)$ , and any $\gamma \in Hom_{N(U)}(F(x),F(y))$ , there exists a covering family $(V_i\to U)$ of $U$ and morphisms $\alpha_i \in Hom_{M(V_i)}(x|_{V_i},y|_{V_i})$ such that $F(\alpha_i) = \gamma|_{V_i}$ for all $i$ . (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 $U$ and any $z\in N(U)$ , there exists a covering family $(V_i\to U)$ of $U$ and $x_i \in M(V_i)$ such that $F(x_i) \cong z|_{V_i}$ .

Now the “constant sheaf” functor $\Delta \colon Set \to Sh_1(C)$ induces a morphism of 2-sheaves from the constant 2-sheaf $\Delta Set$ to the 2-sheaf $\underline{Sh_1(C)}$ 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 $Set$ to $\underline{Sh_1(C)}$ , 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:

$\Delta Set \to LocConst_{\frac{3}{2}} \to LocConst_1 \to \underline{Sh_1(C)}.$

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

So what is the 2-sheaf I called $LocConst_{\frac{3}{2}}$ ? In the familiar classical world, it’s the same as $\Delta Set$ —in other words, the morphism $\Delta Set \to \underline{Sh_1(C)}$ is already faithful. The question here is about whether the map $\Delta Aut(A) \to Aut(\Delta A)$ is monic, which will be true whenever $\Delta$ is sub-cartesian-closed (which, by definition, means that each map $\Delta(Y^X) \to (\Delta Y)^{\Delta X}$ is monic). This is mostly what it means to say that $(\Delta\dashv \Gamma)$ 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 $Set$ , then $LocConst_{\frac{3}{2}}$ might be different from $\Delta Set$ and would give us a separate “Definition $\frac{3}{2}$ ” 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 $(n+2)$ -ary factorization system on the $(n+1)$ -category of $(n+1)$ -sheaves which breaks down the map $\Delta (n-1) Gpd \to \underline{Sh_n(C)}$ into $(n+2)$ potentially different notions of “locally constant $n$ -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 $\underline{Sh_n(C)}$ . 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

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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 $U\to 1$ he studies mainly the topos of sheaves which are constant over $U$ , 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

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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 $G$ on a set $A$ .

A progroup homomorphism $G\to Aut(A)$ , where $Aut(A)$ is regarded as a constant progroup. This is equivalent to regarding $G$ as a localic group and $Aut(A)$ as a discrete group and asking for a homomorphism of localic groups.
A map $G\times A \to A$ of pro-sets which is associative and unital. This is equivalent to asking for a continuous map $G\times A\to A$ of locales which is likewise associative and unital, which in turn is equivalent to asking for a homomorphism $G\to Aut(A)$ of localic groups, where now $Aut(A) \subseteq A^A$ has the product topology. Note that the product topology on $Aut(A)$ will not be prodiscrete in general.

These are the same when $A$ 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 $G$ be the profinite completion of the integers, i.e. the progroup $\underset{\leftarrow}{\lim} \mathbb{Z}/n$ . A homomorphism from $G$ to the constant progroup $Aut(A)$ is just an action of $\mathbb{Z}/n$ on $A$ for some $n$ . However, when $Aut(A)$ has the product topology, then a continuous homomorphism $G\to Aut(A)$ is a $\mathbb{Z}$ -action on $A$ (i.e. an automorphism of $A$ ) 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 $\pi_1(E)$ , 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 $Core(Set)$ .

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

Still thinking…

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

Re: Locally Constant Sheaves

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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 $G_U$ whose objects are “covering projections split by $U$ ,” for any covering family $U$ in a topos $E$ , which is “atomic” and equivalent to the topos of actions of a localic groupoid $\mathbf{G}_U$ . Then he defines a topos $G(E)$ as the limit, in $Topos$ , of the topoi $G_U$ over refinements of covers, and proves that it is equivalent to the topos of actions of a localic progroupoid, which he calls $\pi_1(E)$ . 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

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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 $U\to 1$ , and he studies locally constant sheaves which have a nice splitting over $U$ . But we could look at a slightly different question: given a locally constant sheaf $X$ , it might happen that there exists a covering $U\to 1$ such that $X$ becomes constant over $U$ in the sense of definition 1, but not in the sense of definition 2. However, we might hope that there exists another covering $V\to 1$ such that $X$ becomes trivial over $V$ 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 $X$ such that there exists a decomposition $1=\amalg_i U_i$ , and, for each index $i$ , a group $G_i$ as well as a $G_i$ -torsor $Y_i$ over $U_i$ such that $X$ becomes constant over $Y_i$ . 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 $LC(B \pi_1(T))\simeq LC(T)$ . In other words, for any locally constant sheaf $X$ in this galoisian sense, there exists a geometric morphism $f:T\to B G$ , for $G$ a (discrete) groupoid, such that $X$ is in the (essential) image of $f^*$ : 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

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Dropping back down a few comment nesting levels…

Definition 1’: a sheaf $X$ is locally constant if there exists a decomposition $1=\amalg_i U_i$ as well as epimorphisms $V_i\to U_i$ , some sets $F_i$ , and some isomorphisms $V_i\times X\simeq V_i\times \Delta F_i$ over each $V_i$ .

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 $Y$ be the sheaf of isomorphisms from $\Delta S$ to $X$ . We know that $Y$ admits a section $s$ over $U$ . Define a presheaf $Y_s$ on our ambient topos $T$ as follows. For a sheaf $V$ on $T$ , $Y_s(V)$ is the subset of sections $t$ of $Y(V)$ such that there exists a covering $f: W\to U\times V$ as well as a bijection $g:S\to S$ , such that the map $Y(V)\to Y(U\times V) \to Y(W)$ sends $t$ to the image of $s g$ by the map $Y(U) \to Y(U\times V) \to Y(W)$ .

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

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

However, I still don’t understand why $Y_s$ is a torsor. In fact, it seems to me that if $Y_s$ is well-supported, then $X$ satisfies Definition 2. For suppose that $V$ is well-supported and $Y_s$ has a section $t$ over $V$ . Then $t\in Aut(\Delta S)(V)$ and we have a cover $f\colon W \to U\times V$ and a $g\in (\Delta Aut(S))(W)$ such that $f^\ast(\pi_2^\ast(t)) = f^\ast(\pi_1^\ast(s)) . g$ .

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

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 $Y_s$ is an $\Delta Aut(S)$ -torsor. Indeed, this is a local condition, so that, by replacing $X$ by $Y\times X$ and $T$ by $T/Y$ , we may assume that $X=\Delta S$ and that $s$ is the identity of $X$ . In this case, we have $Y_s=\Delta Aut(S)$ , which is the universal $\Delta Aut(S)$ -torsor.

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

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

Re: Locally Constant Sheaves

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

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

Re: Locally Constant Sheaves

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If $s$ is an isomorphism $U\times X\simeq V\times \Delta S$ over $V$ , as being an $Aut(S)$ -torsor is a local property, to prove that $Y_s$ is a torsor, we may assume that $s$ is given by a global isomorphism $s:\Delta S\simeq X$ (by replacing $T$ by $T/V$ …)

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

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

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

Re: Locally Constant Sheaves

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I had time to think about this a little more, and you are right (I couldn’t find a way to really understand this $Y_s$ …). 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 $G$ as well as a geometric morphism $f:T\to BG$ such that, for any representable object $Y$ in $BG$ , $X$ is constant over $f^* Y$ . 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 $X$ is a $\Delta Aut(F)$ -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 $LC(T)$ as a full subcategory of $T$ , then we will have a canonical equivalence $LC(B\pi_1(T))\simeq LC(T)$ (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 $T$ in general (unless $T$ is locally $0$ -connected). They will form a stack over $B\pi_1(T)$ . 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

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this means that locally constant sheaves (considered as a full subcategory of sheaves) do not form a stack on $T$ 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 $\Delta_T Core(Set)$ on the core of $Set$ . Moreover, if $Core(LC(T))$ denotes the groupoid $Hom_T(1,\Delta_T Core(Set))$ of locally constant sheaves (the full category of locally constant sheaves would be the sections of the constant 2-sheaf $\Delta Set$ rather than the (2,1)-sheaf $\Delta Core(Set)$ ), then we tautologically have

$Core(LC(T)) = Hom_T(1,\Delta_T Core(Set)) = Hom_{Topos}(T, B Core(Set)) = Hom_{pro(Gpd)}(\pi_1(T), Core(Set))$

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

$Core(LC(T)) = ... = Hom_{Topos}(B \pi_1(T), B Core(Set)) = Hom_{B \pi_1(T)}(1, \Delta Core(Set)) = Core(LC(B \pi_1(T))).$

so that at least on cores, the equivalence between locally constant sheaves on $T$ and on $\pi_1(T)$ 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

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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 $k$ , the topos of sheaves on the small étale site of $k$ IS the galois group of $k$ (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 $k$ define a generating family (which just means that any finite separable extension of $k$ is contained in a Galois extension), and apply a general recognition theorem. This is also related with the fully faithfulness of $B$ , which seems to hold: if $B \pi_1(T)$ is the category of sheaves on the full subcategory of locally constant sheaves on $T$ , this implies the fully faithfulness of $B$ .

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

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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 $B$ can possibly be fully faithful on all of Pro(Groupoids), since it is not pseudomonic on Pro(Sets). The pro-set $\cdots \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N} \xrightarrow{+1} \mathbb{N}$ 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 $\cdots \to Set^{\mathbb{N}} \to Set^{\mathbb{N}} \to Set^{\mathbb{N}}$ ) is trivial, i.e. equivalent to $Sh(\emptyset)$ : a geometric morphism $\mathcal{E} \to Set^{\mathbb{N}}$ is a decomposition $1 = \coprod_{i\in \mathbb{N}} U_i$ in $\mathcal{E}$ , and saying that it factors through all the $+1$ s implies that $U_i=0$ for all $i$ , hence $1=0$ 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

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In case anyone is still paying attention to this thread, I think I have found some pro-sets with surjective transition maps for which $B:Pro(Set) \to Locales$ 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

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

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 $\pi_1(T)$ 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

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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 $X$ in the sense of definition 1, there exists a covering $U\to 1$ such that $X$ has a nice trivilisation: the reason for this is that this is true for any object of $B G$ for any discrete groupoid $G$ , and that any locally constant sheaf comes from $B G$ for some $G$ , because of the equivalence $LC(T)=LC(B \pi_1(T))$ . 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 $\pi_1$ for locally connected topoi, the $\pi_1$ of Dubuc leads after all a pro-groupoid which represents torsors in our given topos. The definition of the $\pi_1$ of a topos $T$ I want is the classifying topos of a pro-groupoid which represents torsors in $T$ , 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 $\pi_1(T)$ 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 $T$ for the canonical topology.

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

Theorem I. Let $T$ be a Grothendieck topos. Then $T$ is the classifying topos of a progroupoid if and only if there exists a decomposition $1=\amalg_i U_i$ in $T$ , and for each index $i$ , a group $G_i$ as well as a $G_i$ -torsor $Y_i$ over $U_i$ , such that $\{Y_i \, | \, i\in I\}$ form a generating family of $T$ .

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 $T\to T'$ of Grothendieck topoi iff it induces an equivalence of categories $LC(T')\simeq LC(T)$ .

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

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I have a reason to believe (because of the sketched proof) that, for any locally constant sheaf $X$ in the sense of definition 1, there exists a covering $U\to 1$ such that $X$ 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 $\pi_1(X)$ . It seems to me that a theorem like Dubuc’s about classifying “covering projections” by actions of a progroupoid $\pi_1$ 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

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

Let $X$ be a locally constant sheaf (in the sense of definition 1). Then there exists a (non necessarily unique) decomposition of the terminal sheaf $1=\coprod_i U_i$ and, for each $i$ a set $A_i$ , such that $X\times U_i$ is locally isomorphic to $U_i\times\Delta A_i$ for each $i$ .

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 $\coprod_j W_j \twoheadrightarrow 1$ with $X\times W_j \cong X\times \Delta B_j$ for each $j$ and some sets $B_j$ . Define $j\sim j'$ if $B_j \cong B_{j'}$ . Clearly this is an equivalence relation; let the $i$ ’s be its equivalence classes, let $U_i$ be the support of $\coprod_{j\in i} W_j$ , and define $A_i = B_j$ for any $j\in i$ (of course it doesn’t matter which). Then $X\times U_i$ is locally isomorphic to $A_i$ (over the cover $\coprod_{j\in i} W_j$ ), and we clearly have $\coprod_i U_i \twoheadrightarrow 1$ , so it remains to show that the $U_i$ are disjoint.

But $U_i \cap U_{i'}$ is covered by $\coprod_{j\in i, j'\in i'} W_j \times W_{j'}$ , and $X\times W_j \times W_{j'}$ is isomorphic to both $\Delta B_j\times W_j \times W_{j'}$ and $\Delta B_{j'}\times W_j \times W_{j'}$ . Thus, if (the slice topos over) $W_j\times W_{j'}$ has a point, then $B_j \cong B_{j'}$ . Conversely, therefore, if $i\neq i'$ , then $B_j \ncong B_{j'}$ , so $W_j\times W_{j'}$ has no points. If our topos has enough points, this implies that $W_j\times W_{j'}$ is initial, and hence so is $U_i\cap U_{i'}$ .

In general, however, I’m pretty sure it can happen that $\Delta A \cong \Delta B$ in a nontrivial topos even though $A\ncong B$ . For instance, we could consider the classifying topos of bijections between $A$ and $B$ . Thus, it seems that in general we could have a covering $U \sqcup V \twoheadrightarrow 1$ such that $X\times U \cong \Delta A \times U$ and $X\times V \cong \Delta B \times V$ , but $U\cap V \neq 0$ and $A\ncong B$ , 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

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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 $X$ , 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. 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 $\amalg_j W_j \to 1$ as well as a trivialisation $X\times W_j\simeq \Delta B_j\times W_j$ over $W_j$ for each $j$ . Then define a relation on indices by $j\sim j'$ if $\Delta B_j\simeq \Delta B_{j'}$ . This is an equivalence relation. Denote by $[j]$ the equivalence class of $j$ . Define $V_{[j]}=\amalg_{j'\in [j]}W_{j'}$ , and define $B_{[j]}=B_j$ . I claim that, if $i$ and $j$ are not equivalent, than $V_{[i]}\times V_{[j]}$ is empty. To see this, we choose a boolean cover of $T$ , 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

$X\times V_{[j]} \simeq \Delta B_j\times V_{[j]}$

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

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

$U_i\times \Delta S_i\simeq U_i\times X$

over $U_i$ for all $i$ , with the following property: for indices $i$ and $j$ , if $i\neq j$ , then $U_i\times U_j$ is empty. Therefore, we must have

$1=\amalg_i U_i_, .$

Now, consider a locally constant sheaf $X$ , and assume for simplicity that there exists a set $S$ as well as a covering of shape $U\to 1$ such that $U\times X\simeq \Delta S\times U$ over $U$ . Let $Y$ be the sheaf of isomorphisms from $\Delta S$ to $X$ . We know that $Y$ admits a section $s$ over $U$ . Define a presheaf $Y_s$ on our ambient topos $T$ as follows. For a sheaf $V$ on $T$ , $Y_s(V)$ is the subset of sections $t$ of $Y(V)$ such that there exists a covering $f: W\to U\times V$ as well as a bijection $g:S\to S$ , such that the map

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

sends $t$ to the image of $sg$ by the map

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

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

$\Delta Aut(S)\to Aut(\Delta S)$

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

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

So we have proved what I claimed from the begining: for any locally constant sheaf $X$ , there exists a decomposition $1=\amalg_i U_i$ , a family of sets $S_i$ , a family of groups $G_i$ , as well as, for each index $i$ a $G_i$ -torsor $Y_i$ over $U_i$ , such that $X$ is constant over $Y_i$ .

This may be reformulated as follows.

Given a locally constant sheaf $X$ on a topos $T$ , there exists a (discrete) groupoid $G$ and morphism of topoi

$f:T\to BG=\hat{G}$

such that, for any representable presheaf $Y$ over $G$ , $X$ is constant in $T/f^*Y$ .

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

$LC(T)\simeq LC(B\pi_1(T))$

for any topos $T$ . The proof of this equivalence uses nothing than the existence of $\pi_1(T)$ (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 $T$ , the following conditions are equivalent. (i) $T$ is generated by its objects of shape $Y$ , such that there exists a decomposition $1=U\amalg V$ as well as a group $G$ , such that $Y\times V=\empty$ and such that $Y$ is a $G$ -torsor over $U$ ; (ii) $T$ is generated by a family locally constant sheaves; (iii) $T$ 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 $LC(T)=LC(B\pi_1(T))$ is a site of definition of $T$ , so that $T=B\pi_1(T)$ .

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 $k$ 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. 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

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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 $S^1\times \{\ell\}$ 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

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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

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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

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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 $1=\amalg_i U_i$ 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 $k$ admits a root in a finite Galois extension of $k$ ):

Theorem. 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 $f:T\to BG$ such that, for any representable presheaf $Y$ in $BG$ , $X$ becomes constant over $f^*Y$ .

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

Consider a morphism of topoi $p:T\to S$ as well as a sheaf $F$ over $S$ . Say that a sheaf $X$ over $T$ is locally isomorphic to $F$ is there exists a covering $U\to 1$ of $T$ as well as an isomorphism $U\times X\simeq U\times p^* F$ over $U$ . We may always assume that there exists a sheaf of groups $G$ over $S$ as well as a $p^*(G)$ -torsor $Y$ in $T$ , such that $Y\times X\simeq Y\times p^* F$ over $Y$ . To prove this, we pick a trivilisation $s:U\times X\simeq U\times p^* F$ , we put $G=Aut(F)$ , and we define $Y$ to be the $p^*(G)$ -torsor of isomorphisms from $p^*(F)$ to $X$ which are locally in the orbit of $s$ .

We may now prove the theorem as follows. Consider a set $I$ , a covering of shape $\amalg_{i\in I} U_i\to 1$ , a family of sets $\{F_i\}_{i\in I}$ , as well as a family of isomorphisms $U_i\times X\simeq U_i\times \Delta F_i$ over $U_i$ . Consider the natural projection

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

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

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

Re: Locally Constant Sheaves

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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 $f^\ast Y$ .

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

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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 $X$ is locally constant if there exists a decomposition $1=\amalg_i U_i$ as well as epimorphisms $V_i\to U_i$ , some sets $F_i$ , and some isomorphisms $V_i\times X\simeq V_i\times \Delta F_i$ over each $V_i$ .

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

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

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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 $n$ -sheaf $\Delta core((n-1)Grpd_\kappa)$ is the $n$ -sheaf of locally constant $\kappa$ -bounded $(n-1)$ -sheaves.

Def 2 is just unwinding what the sections of the constant $n$ -sheaf $\Delta core((n-1) Grpd_\kappa)$ are.

I’d think.

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

Re: Locally Constant Sheaves

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

we have

$core (n-1)Grpd_\kappa \simeq \coprod_{i} \mathbf{B}Aut(A_i)$

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

$\Delta core(n-1)Grpd \simeq \coprod_i \mathbf{B}\Delta Aut(A_i) \,.$

Now, every connected object $\mathbf{B}G$ in an $n$ -topos classifies $G$ -principal $n$ -bundles.

The canonical $n$ -permutation representation

$\rho : \mathbf{B} Aut(A_i) \to (n-1)Grpd$

induces a notion of $\rho$ -associated $n$ -bundles.

One can see that def 2 defines a locally constant $(n-1)$ -sheaf to be, on each connected component, the $\rho$ -associated $n$ -bundle of an $\Delta Aut(A_i)$ -principal $n$ -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

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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

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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

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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

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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 $n$ Lab 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

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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

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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

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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

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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

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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

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Let’s see, I think I remember reading that the etale topos of a field $k$ is the classifying topos for separable closures of $k$ . 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 $Set^P \to Sh(X_{et})$ for $P$ the set of points. Since $Sh(X_{et}) \to Set^{\pi_1(X)}$ is also supposed to be surjective (in fact, connected), we should have a surjection $Set^P \to Set^{\pi_1(X)}$ , which should allow us to regard $\pi_1(X)$ as a (pro)groupoid with $P$ as its set of objects. But if $Sh(X_{et})$ is itself connected, then the groupoid $\pi_1(X)$ 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

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

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

Example

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

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

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

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

Re: Locally Constant Sheaves

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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 $k$ is, I read, just a finite disjoint union of Specs of finite separable extensions of $k$ .

I guess that given any étale cover $Y\to X$ , 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 $k$ . I’m going to speculate that the separable closure determines another étale sheaf on $Spec(k)$ , 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 $k$ is locally connected, that should imply that the (profinite) fundamental groupoid can be identified with (the profinite completion of) the image under $\Pi_0 \colon Sh(k_{et}) \to Set$ 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

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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 $n$ Lab: 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

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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

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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

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In what sense do you mean the “classifying topos” of $U(1)$ ? Since $U(1)$ is connected, it cannot act nontrivially on a discrete set, so the category of continuous $U(1)$ -sets is just $Set$ , whose fundamental groupoid is trivial. It probably has a nontrivial classifying $(\infty,1)$ -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

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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 $U=(U_i)$ and the category $\mathcal{P}_U$ of “locally constant sheaves (Def. 1) trivialized by $U$ ”, with a subcategory $\mathcal{G}_U$ of the “covering projections (Def. 2)”. Then he identifies $\mathcal{G}_U$ with the classifying topos of a localic groupoid, and then takes the limit over refinements of $U$ to get a pro-localic groupoid.

However, I would be inclined to further break down the categorgies $\mathcal{G}_U$ , by replacing the covering family $(U_i)$ by a 1-bounded hypercover, i.e. in addition to $(U_i)$ we include as data a covering $(V_{i j k})$ of each $U_i \cap U_j$ . Then a locally constant object (Def. 2) is “trivialized over $(U,V)$ ” if it becomes constant over each $U_i$ , and the transition functions from $U_i$ to $U_j$ become constant over each $V_{i j k}$ . (Rather than merely, as in Dubuc’s definition, asking for $V_{i j k}$ 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

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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

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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 $(2,1)$ -toposes we might assume local $1$ -connectivity. And for a Galois $(\infty,1)$ -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 $(2,1)$ -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 $(\infty,1)$ -toposes, but conversely in finding as many good big locally $\infty$ -connected $(\infty,1)$ -toposes as possible. A class of examples is provided by $\infty$ -stacks over $(\infty,1)$ -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

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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

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When discussing the fundamental ∞-groupoid of a topos, I would prefer to point people to fundamental $\infty$ -groupoid of a locally $\infty$ -connected $(\infty,1)$ -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

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I am in the process of adding at $n$ Lab: 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

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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

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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 $k$ , 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 $k$ . It seems to me that once we have these facts, the correspondence between field extensions and $Gal(k)$ -actions requires only knowing that locally constant sheaves in a topos $E$ are classified by functors $\pi_1(E)\to Set$ . Moreover, this follows tautologically from Definition 2 of locally constant sheaves and a definition of $\pi_1$ 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 $G$ 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 $Sh(k_{et})$ is equivalent to the classifying topos of $\pi_1(Sh(k_{et}))$ , 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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