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I have touched étale groupoid and various entries related to this.
I have made orbit space redirect to orbit, though eventually it might want to be a separate entry.
Also I have made foliation theory redirect to folitation, though eventually it might want to be a separate entry.
I have added DeligneMumford stack as a “related concept” to étale groupoid, though eventually what I am after is a complex of entries that discusses approaches to a general notion of étale ∞groupoids and how these subentries fit into a more general story.
So I’ll be creating a stub étale ∞groupoid, but I am not sure if I have time and energy to have it be more than a reminder for things to look into later.
I think of them, too. My first idea was to work with the description of the source and target
The source of a $k$cell $\phi\in C([k])$ is defined to be the pasting[^pasting] (or composite) of $d_k\phi$, $d_{k2}\phi$, $d_{k4}\phi$,…
The target of a $k$cell $\phi\in C([k])$ is defined to be the pasting (or composite) of $d_{k1}\phi$, $d_{k3}\phi,$ $d_{k5}\phi$,…
of a ncell and require them to be étale morphisms for an appropriate notion of ”étale” which at least should include the condition to be formally étale.
To Urs: I wrote to David Garchedi and asked him if he is interested in discussing ∞orbifolds with me  but unfortunately got no response. Do you know if the contact on his website is up to date?
I think of them, too.
I know, you told me. And so I was getting a tad worried concerning your project when I heard David announce all this progress. But, on the other hand, there should be plenty of fruit to be picked here. In any case, I think you should stay in contact about these developments.
Do you know if the contact on his website is up to date?
I know that it is no longer up to date. Instead, you can reach him as Google user “davidcarchedi”.
The source of a $k$cell $\phi\in C([k])$ is defined to be the pasting (or composite) of $d_k\phi$, $d_{k2}\phi$, $d_{k4}\phi$,…
You are trying to turn it globular, it seems. Is this necessary? Dave claims that the right condition is simply that all face maps are étale (for the degeneracies it should then follow).
Hm, of course it is great to have these topics treated so extensively but for me it is very sad to write a now rather superfluous work (ok, ”stable” and ”proper” are left, but I guess ”proper” is described in Luries ”Higher algebra”).
In my condition above, degeneracies do not appear and I don’ t know what he has in mind with them. Also intuitively for me source and target ”are faces and not degeneracies”. I will have to think about the statements that (1) étaleness of the faces implies that of degeneracies and (2) étaleness of the pasted diagram is equivalent to étaleness of the summands also in higher degrees. I guess (1) if it is true should be just a consequence of the simplicial identities.
Perhaps it is also in the case of étaleness good to not only look at the definition of étale, but also what follows from it. In the case of properness it took me a bit to realize, that this condition is conceptually better seen as two different requirements  namely stability (local finiteness of the action) and ruling out the branching…
Thanks Urs, for your concern. I will see what is left for me to do now. I regret that I didn’ t get into contact with David Carchedi earlier.
for me it is very sad to write a now rather superfluous work
I was afraid you would say that.
I always tend to advise to the contrary: math is a winwin game, and one person’s progress is the other person’s springboard. You could also say: wow, my thesis is cutting edge research. That’s something. Many a thesis never gets close to this. And ordinary étale groupoid theory is a fundamental and big topic, so higher étalegroupoids will be even more fundamental and even bigger. There is room there for more than one researcher, to be sure.
I think you should press on. But of course you should check with Dave and with your advisor. Dave is a very nice guy, by the way.
In my condition above, degeneracies do not appear
The condition on the degeneracies should be automatic from that on the faces. This is as for ordinary étale groupoids: just requiring the source (which is one of the face maps) to be étale already implies that also the identityassigning map (which is one of the degeneracies) is étale: this is because sections of étale maps are étale (so, yes, this is implied by the simplicial identities).
I haven’t thought about the higher analogs for properness at all yet.
There is room there for more than one researcher, to be sure.
For instance, one central aspect that is still open is an intrinsic characterization of higher étale stacks (as objects in the ambient gros $\infty$topos, at least).
I have the feeling it might be sufficient to flesh out this simple observation a bit, to get both a nice general theory and a proof that the explicit constructions by étale simplicial objects is a presentation for it.
Maybe something you’d enjoy looking into in this context?
Or maybe it turns out that this idea is not good enough. Then one can try somerthing else…
Thanks for your encouragement, Urs. I believe in the cooperational dimension of mathematics (and science in general), too, and looked contemptuous at those not thinking this way as people just encumbering progress; only today for a few moments feelings of some entirely different kind cropped up…
I had glanced at this approach a few weeks ago with the intention to write it down in terms of the en vogue homotopy type theory but laid it aside then. But I guess now it may be time to reactivate the idea.
For instance, one central aspect that is still open is an intrinsic characterization of higher étale stacks (as objects in the ambient gros $\infty$topos, at least).
Maybe something you’d enjoy looking into in this context?
I had glanced at this approach a few weeks ago with the intention to write it down in terms of the en vogue homotopy type theory but laid it aside then. But I guess now it may be time to reactivate the idea.
I like your idea (and the big topos perspective in general) and the ”simple observation” seems to generalize to the higher case, so I will look at this. If there is something other than this you always wanted to know and feel like having me to think over, feel free to let me know it.
Hi Stephan,
that sounds good!
the ”simple observation” seems to generalize to the higher case, so I will look at this. If there is something other than this you always wanted to know and feel like having me to think over, feel free to let me know it.
Yes, I was hoping you would say something like this :)
So here is what I am imagining:
this “simple observation” and some thoughts that follow are something like a sketch of what I guess might be a proof that étale simplicial manifolds present an “intrinsic” definition of étale $\infty$groupoids in Smooth∞Grpd.
I don’t have the leisure at the moment to follow this through in detail. And doing so will lead to lots of further ideas for what to do, and I don’t have the leisure for that, either.
So I wouldn’t mind “offering” this to you, and you can see if you can use this sketch to do something with in your thesis.
First step would be: make that “simple observation” into a formal proof. See how it generalizes (for instance I mentioned manifolds instead of general topological spaces there, but I think this is not really necessary.)
Then second step would be: show that locally constant $\infty$stacks (so: “covering $\infty$bundles”) over a manifold have a presentation by “locally constant étale simplicial manifolds” (those for which the étale structure maps are in fact covering spaces for each connected component). This should be straightforward using the fact that smooth manifolds is an infinitycohesive site. I can give more details here.
Then the third step would be: put the pieces together and characterize those morphisms of simplicial presheaves $E \to X$ that receive an effective epimorphism from a “locally constant étale simplicial manifold”. The proposal would be that these are the presentations of étale $\infty$stacks. Some steps here look obvious, but what might require more thinking it to get the right homotopycorrect characterization. I haven’t really thought about this yet, but we can talk about it.
Alternatively, if somewhere along the way something turns out not to work as nice as it should: think about if there is a way to modify the original “simple observation” to some other characterization of étale spaces by covering spaces (and hence then of étale $\infty$stacks by locally constant $\infty$stacks).
Once at step three, there is a myriad of exciting routes to proceed. Notably at that point one would want to make contact with whatever theory Dave has made stabilize (that should be more details on the presentation side, to which the above would add the general abstract perspective), and combine to a total picture, which will be bigger than the two pieces that it consists of. But maybe this should be thought about then when the time is ripe.
Unless I am overlooking something, this looks like a plan that starts out with something very down to earth and has a continuous tractable road map to something interesting and eventiually to something very interesting.
Might be worth a try.
Hi Urs,
your offering is compelling and I gratefully accept it!
I already sent it to David to ensure that he hasn’t this issue, too, prefabricated in his records.
Well, he says he might or might not have it.
Nevertheless I may adopt your program and already looked at the the ”simple observation” in Top: By definition of a sheaf of sections, given such a cover the right vertical arrow is a ”locally split epi”  for it being locally mono one might need further assumptions.
he says he might or might not have it.
Let’s see, you may have to remind me what “have it” refers to.
the right vertical arrow is a ”locally split epi”
Is this about the étale map associated to a sheaf? Why is that locally epi in general? Consider, in the extreme case, the initial sheaf, which assigns the empty set everywhere.
Let’s see, you may have to remind me what “have it” refers to.
I referred to an ”intrinsic” definition of étale ∞groupoids. But he said that he tried it in a way different to that you describe above.
Is this about the étale map associated to a sheaf? Why is that locally epi in general? Consider, in the extreme case, the initial sheaf, which assigns the empty set everywhere.
What I meant is the following: Given the canonical diagram
$\array{ U_i \times \Gamma_X(Q)(U_i) &\to& Q \\ \downarrow && \downarrow \\ U_i &\hookrightarrow& X } \,,$coming with the locally constant sheaf on $X$ and figurating in the ”cover” of the map $Q\to X$ obtained by taking the coproduct over all open $U\subset X$. We expect then $Q\to X$ to be étale (a local homeomorphism in Top). Now given some $U$ of the cover, the sheaf of local sections gives the set $\Gamma(p)(U):=\{s:U\to Q_{_U}  p_{_U}\circ s =id_{U}\}$ which we assume to be nonempty (at least every point of $Q$ should lie in an open set with nonempty fiber). Hence $p_{_U}$ has a section. So since a monic split epimorphism is always an isomorphism this would prove the ”simple observation” . But maybe I am missing something in the setup since I don’ t see $p_{_U}$ being monic.
Hi,
let’s see. You say:
… coming with the locally constant sheaf on $X$…
So the sheaf on $X$ here is not necessarily locally constant. The idea is that on $X$ we have any sheaf, hence $p : Q \to X$ any étale space. If $X$ happens to be locally constant, then $p : Q \to X$ happens to have the same fibers on every connected component of $X$. But in general the fibers of $Q$ may vary very much. In particular they may be empty.
A good example to keep in mind for visualization is the étale space given by a collection of open subsets $\{K_i \to X\}$ (not necessarily covering $X$). Take $Q := \coprod_i K_i$ to be their disjoint union. Equipped with the evident projection $p : \coprod_i K_i \to X$ this is an étale space.
For this example it is easy to say what the local sections of $p$ are. Over any $U \hookrightarrow X$, the set of sections is the subset
$\Gamma_X(Q)(U) \subset \{K_i\}$on those $K_i$ such that $U \subset K_i$. There may be many such $K_i$, a single one, or none.
So the étale map $p : Q \to X$ will in general neither be (locally) epi nor (locally) mono.
The idea with the relation to covering spaces is this: suppose, in the above example, we find a $U$ such that all the $K_i$ that it intersects already entirely include it. Then this means that restricted to this $U$, the map $p_U : Q_U \to U$ does have constant fibers now: the fiber over every point of $U$ is identified with the same set of points (identified with the set of indices $i$ such that $U$ is in $K_i$). So over such $U$ the restriction of $p$ now is in fact a covering space. The restriction $p_U : Q_U \to U$ is the étale space coresponding to the constant sheaf on $U$ which to every subset of $U$ assigns this same set.
In general $p : Q \to X$ will not be such that we can find a cover $\{U_i \to X\}$ such that all the $U_i$ have precisely the above property (in the above exampe, a $U_i$ might intersect some $K_i$ without being included in it). But we can slightly modify this construction, and still simply consider for each $U_i$ the set of sections of $p : Q \to X$ over it. And we should still get a cover of the étale space $Q \to X$ by a bunch of covering spaces.
I suspect this discussion may be within the expertise and intersecting with the subject of some recent papers of Marta Bunge (in 1categorical context).
some recent papers
Which ones do you have in mind?
I don’t know., she mentions the similar words in conference abstracts like covering morphisms in topos context etc. Also she always took internal points of view on things in stack theory. I know that Igor was doing 2etale spaces and compared some of the stuff to her earlier papers…
Hi Urs,
there have apparently been two misunderstanding (what you explained thereafter was in principle entirely clear to me before):
… coming with the locally constant sheaf on X …
I meant $Q$  so, sorry! In particular the left morphism in the below diagram is a covering space / the étale morphism corresponding to a locally constant sheaf
The idea is that ….
It is: ”A map $p:Q\to X$ of topological spaces is étale iff it is covered by a trivial covering space” (where just to have the diagram here  ”covered by a trivial covering space” meant to have the canonical commuting diagram
$\array{ \coprod_i ( U_i \times \Gamma_X(Q)(U_i)) &\to& Q \\ \downarrow && \downarrow \\ \coprod_i U_i &\to& X } \,.$)
So the étale map $p:Q\to X$ will in general neither be (locally) epi nor (locally) mono.
Of course $p$ is neither epi nor mono in general, but I speak ”étale map” if it is locally an isomorphism. I tried then to show the implication from right to left above (”covered by a trivial covering space” implies ”étale”) by arguing that there are appropriate restrictions of $p$ which are split epimorphisms (and monomorphisms). I shouldn’t have called this ”local epimorphism” etc. since this terminology is reserved for some morphism out of a system of local epimorphisms.
I hope you excuse this bit of confusion.
I will have a look at Marta Bunge.
Oh, I see. I am sorry for misunderstanding you. Okay, good.
So here is how I would imagine to argue that given the diagram as above, the right map is étale:
Pick any point $q$ in $Q$. By assumption it is in the image of some $U_i \times \{\sigma\}$ under the top horizontal morphism. Let me call that image set $\sigma(U_i)$, even though that notation is suggesting more than we strictly know at this point.
Then by commutativity of the diagram, we know that, under $p$, $\sigma(U_i)$ maps to $U_i$. Since this is open and $p$ is continuous, it follows that the neighbourhood $\sigma(U_i) \subset Q$ is open in $Q$.
So for deducing that $p$ is étale it would now be sufficient to argue that $p$ restricts on this $\sigma(U_i)$ to a homeomorphism $p_{\sigma(U_i)} : \sigma(U_i) \simeq U_i$.
It is clear that this is a split epimorphism, by commutativity of the diagram and the assumption on the left morphism. So the question remains why or indeed if $p_{\sigma(U_i)}$ is a monomorphism.
But that’s exactly what you have been saying! Okay, so I am following you now. :) Sorry again.
Hm, but when we restrict the diagram to $U_i \times \{\sigma\}$ in the top left corner, don’t we get the diagram
$\array{ U_i \times \{\sigma\} &\stackrel{epi}{\to}& \sigma(U_i) \\ {}^{\mathllap{id}}\downarrow &\searrow^{\mathrlap{id}}& \downarrow^{\mathrlap{p_{\sigma(U_i)}}}_{epi} \\ U_i &\stackrel{id}{\to}& U_i } \,.$Which seems to imply that the right morphism is an iso?
I will have a look at Marta Bunge.
I have seen a few of their articles. But none that seemed to be relevant to the particular question we are discussing here. But I haven’t seen all of them, maybe.
Yes, $\sigma$ is an isomorphism (since as a section it is split mono and it is epi since $h=\sigma\circ \pr$ is epi
$\array{ U_i \times \{\sigma\} &\stackrel{h}{\to}& \sigma(U_i) \\ {}^{\mathllap{\pr}}\downarrow &\nearrow^{\mathrlap{\sigma}}& \downarrow^{\mathrlap{p_{\sigma(U_i)}}}_{epi} \\ U_i &\stackrel{id}{\to}& U_i } \,.$)
hence its retraction $p_{\sigma(U_i)}$ is iso, too. So, this step is passed thanks!
Since in the proof is only used that one can restrict maps and some reasoning about epis and monos I guess this statement generalizes to other sites.
But following your program I will think next about higher sheaves on the same sites (Top or Mfd), is that ok?
But following your program I will think next about higher sheaves on the same sites (Top or Mfd), is that ok?
Okay, great, let’s think about what to do next.
Maybe let’s pause a moment and sort out what the statement is that we are after.
There seem to be two different concepts here, which are related, but not equivalent:
étale morphisms between $\infty$stacks;
étale $\infty$stacks.
Right? Maybe Dave has understood this more deeply than I have. I was thinking:
an ordinary étale groupoid is a groupoid such that its source map, say, is an étale map. Read: such that its source map is an étale morphism of (smooth) 0groupoids.
If this is the right intrinsic perspective, then we are already done with characterizing étale groupoids abstractly. We just say: are smooth stacks $A$ such that there is an effective epimorphism $U \to A$ out of a 0truncated $U$ such that any of the two projections $U \times_A U \to U$ out of the homotopy fiber product is an étale morphism of smooth 0groupoids (which we have already characterized).
If that’s the right perspective, then the next step would seem to be
to characterize étale morphisms between 1stacks / smooth groupoids in our fashion.
Define an étale 2stack / 2groupoid $A$ to be one such that there is an atlas $U$ etc. such that $U \times_A U \to U$ is an étale morphism of stacks / smooth groupoids.
That’s what I believe I was thinking of. Does that sound reasonable to you? There might be other ways to go about it.
I agree to a large extend on what you said:
There seem to be two different concepts here, which are related, but not equivalent:
étale morphisms between stacks;
étale stacks.
David didn’t mention ”higher étale morphisms” ( étale morphisms between stacks) to me. As I understood him (but I haven’t the notes of his talk on this, yet), his étale ∞stacks are the objects of $\Sh_{(\infty,1)}(\Mfd^\et)$ where $\Mfd^\et$ is the category whose objects are manifolds and the morphisms are étale maps of them. In this picture étale ∞groupoids are then groupoid objects in $\Sh_{(\infty,1)}(\Mfd^\et)$, so he does not need to refer to some étale morphisms between ∞stacks. Maybe he calls the morphisms between étale ∞stacks ”étale”. I guess the major difference between our approach and David’s is that the ”higher étale morphisms” are only implicit in his.
an ordinary étale groupoid is a groupoid such that its source map, say, is an étale map. Read: such that its source map is an étale morphism of (smooth) 0groupoids.
This condition is equivalent to requesting all structure maps of the internal groupoid to be étale. Because of this David started with $\Mfd^\et$ in first place.
If that’s the right perspective, then the next step would seem to be
to characterize étale morphisms between 1stacks / smooth groupoids in our fashion.
Define an étale 2stack / 2groupoid $A$ to be one such that there is an atlas $U$ etc. such that $U \times_A U \to U$ is an étale morphism of stacks / smooth groupoids.
That’s what I believe I was thinking of. Does that sound reasonable to you?
Yes
There might be other ways to go about it.
There is the fundamental theorem of covering spaces which one could bring into play, once the characterization via the covering space is settled.
This condition is equivalent to requesting all structure maps of the internal groupoid to be étale.
That’s true for étale simplicial manifolds and the ordinary notion of étale maps.
To even make sense of this now fully internally we first need to be able to say when we a morphism $A \to B$ of $\infty$stacks is étale in analogy to when a continuous map $X \to Y$ exhibits $X$ as an étale space over $Y$.
Once we have that we can ask:
for $A$ an $\infty$stack and $U \to A$ an effective epimorphism, such that in the corresponding groupoid object
$\cdots U \times_A U \times_A U \stackrel{\to}{\stackrel{\to}{\to}} U \times_A U \stackrel{\to}{\to} U$one of the rightmost morphism is an étale morphism of $\infty$stacks, is it then true that all the other morphisms here are étale morphism of $\infty$stacks.
Or similar questions. Probably it will be true. But this is not clear yet. At least it first requires making some definition.
Of course we are talking about a proposal for that definition: say a morphism $U \times_A U \to U$ of $\infty$stacks is étale if there is a morphism $E \to B$ which is the total space of a locally constant $\infty$stack on $B$ and a diagram of $\infty$stacks
$\array{ E &\to& U \times_A U \\ \downarrow && \downarrow \\ B &\to& U }$with the two horizontal morphisms effective epis.
There is the fundamental theorem of covering spaces which one could bring into play, once the characterization via the covering space is settled.
Yes, exactly. This is precisely where I am coming from here.
So I should expand on this:
the point is that for those ambient $\infty$toposes that I call “cohesive” – which includes the one that one cares about when talking about étale Lie groupoids – we have a good characterization of locally constant $\infty$stacks, hence of “covering morphisms” of $\infty$stacks, including, notably, the $\infty$analog of the “fundamental theorem of covering spaces”.
This is all discussed in section 2.3.13. The “fundamental theorem of $\infty$covering spaces” is prop. 2.3.124 there.
So locally constant $\infty$stacks and covering morphisms of $\infty$stacks are canonically and abstractly given (“godgiven”, it’s not our choice) and if the ambient $\infty$topos is cohesive, then they behave in the way that they should.
So this was my starting point when I started thinking about étale $\infty$stacks. I thought it would be good if these could be defined in terms of this canonical notion that we already have.
So, yes, it’s all about getting the notion of ételness starting from the universal theorem of covering spaces.
I am following
This is all discussed in section 2.3.13. The “fundamental theorem of $\infty$covering spaces” is > prop. 2.3.124 there.
As I understand it in light of
Observation 2.3.130, p.171
For $f : X\to Y$ a $\Pi$closed morphism, its fibers $X_y$ over global points $y : * \to Y$ are discrete objects.
the definition you have in minds reads:
A morphism $f:Q\to X$ in an (∞,1)category is called étale if there is a commuting square
$\array{E&\to&Q\\\downarrow^p&&\downarrow^f\\B&\to&X}$such that
the horizontal arrows are effective epimorphisms
the left arrow is $\Pi$closed and $E$ is locally constant (∞,1)stack on $B$.

Then by
Proposition 2.3.131, 171
Let $H$ have an ∞cohesive site of definition, 2.2.2. Then for any $X \in H$ the locally constant ∞stacks $E \in LConst(X)$, regarded as ∞bundle morphisms $p : E \to X$ by >observation 2.3.123, are precisely the $\Pi$closed morphisms into X.
the codomain of $p$ is $Q$ what we expected from the 0dimensional case $\Sh(Q)\simeq\Et/Q$ and ”$L\Const(B)\simeq\Cov(B)$.
then the next step would seem to be
to characterize étale morphisms between 1stacks / smooth groupoids in our fashion.
Define an étale 2stack / 2groupoid $A$ to be one such that there is an atlas $U$ etc. such that $U \times_A U \to U$ is an étale morphism of stacks / > smooth groupoids.
So this step was already in section 2.3.13 and it restricts to the case n=0  once this step is shown for a general cohesive site. So this would be the next step, I guess?
Yes!
Essentially, up to one minor point. Let’s sort this out: you write:
 the left arrow is $\Pi$closed and $E$ is a locally constant (∞,1)stack on $Q$.
On $B$. I guess this is a typo on your end?
A $\Pi$closed morphism $E \to B$ is to be thought of as the étale space projection of a locally constant $\infty$stack on $B$.
(What can be a bit confusing here – but is the very reason why this is interesting! – is that “stack” appears here in two meaning, an external and an internal one:
First, we are in an $\infty$topos whose objects we may think of as $\infty$stacks on some site.
But next, for a given such object $X$ or $B$ or whatever, we want to say what it would mean to have a stack on that object. A stack on a stack. We want to say:
a morphism of objects in the $\infty$topos $Q \to X$ is like the étale space of an $\infty$stack on the $\infty$stack $X$.)
On $B$. I guess this is a typo on your end?
Yes, I’m afraid it (again) is :(
A $\Pi$closed morphism $E\to B$ is to be thought of as the étale space projection of a locally constant ∞stack on $B$. (What can be a bit confusing here – but is the very reason why this is interesting!…
$E\to B$ is considered as an object in the petit ∞topos $H/B$ over $B$ and called ∞stack on $B$ as usual, I assume. I have just to look again at the identification process for this to make sense.
(What can be a bit confusing here – but is the very reason why this is interesting!…
This is probably how to pass from $Top$ to a general ∞cohesive site $C$. If $E,B\in C$ are objects then there may be a related notion of ”étale”, namely a local isomorphism $p:E\to B$ (considered as ∞presheaves) meaning that between the sheafified $p^':E^'\to B^'$ is an isomorphism in $H:=\Sh_\infty(C)$. So a statement to check would read: ”A morphism in a ∞cohesive site is a local isomorphism iff it is covered by a trivial covering space”
(i.e. in the definition / proposition
(
A morphism $f:Q\to X$ in an (∞,1)category is called étale if there is a commuting square
$\array{E&\to&Q\\\downarrow^p&&\downarrow^f\\B&\to&X}$such that
the horizontal arrows are effective epimorphisms
the left arrow is $\Pi$closed and $E$ is locally constant (∞,1)stack on $B$.)
it is to verify if it is true for $Q$, $X$ etc. in a general ∞cohesive site and ∞stacks on them. I ask this (maybe again) since we didn’ t talk about the statement to be valid in other ∞cohesive sites than Top.)
(This is not related but do you know if the category with manifolds (or topological spaces) as objects and proper maps as morphisms is an ∞cohesive site?)
Hi Stephan,
sorry for the slow reply, but I was all absorbed with this.
And now it’s really too late at night for me to be replying, so I’ll be brief and say more later:
E→B is considered as an object in the petit ∞topos H/B over B and called ∞stack on B as usual, I assume. I have just to look again at the identification process for this to make sense.
Almost yes. So we canonically have the slice $\infty$topos $\mathbf{H}/B$. But that’s big. Too big. We want to identify in there a sub$\infty$category (hopefully a sub$\infty$topos) of those objects that actually shalify as $\infty$sheaves = $\infty$stacks on $B$.
Break this down to the 1categorical case to see what’s happening. For $B$ a smooth manifold, the slice $Smooth\infty Grpd / B$ or even hust the slice $Sh(SmthMfd)/B$ contains lots and lots of things. For instance all smooth manifolds $X$ equipped with smooth maps to $B$ are in there. But very few of these maps $X \to B$ correspon to actual sheaves on $B$. Namely these are only those maps $X \to B$ which are étale. The étale maps form a subcategory
$Et(B) \hookrightarrow \mathbf{H}/B$of the slice, and this we want to identify. By higher analogy with the étale space / sheafduality, we then say that the objects in $Et(B)$ are (the total spaces of) the $\infty$sheaves on $B$.
a related notion of ”étale”, namely a local isomorphism
No. This is maybe what led to misunderstandings earlier: local isomorphisms are not to be identified with local homeomorphisms, despite the similarity of name. That’s the reason why I’d rather keep saying “étale map” instead of “local homeomorphism”. The latte just collides too much as we generalize.
(This is not related but do you know if the category with manifolds (or topological spaces) as objects and proper maps as morphisms is an ∞cohesive site?)
I haven’t thought about this and am lacking intuition right now. I’ll try again later, when I am more awake… ;)
Hi Urs,
sorry for the slow reply
You certainly do not have to apologize! Since you said you have actually no leisure for the subject now, I take it by no means for granted that you comment on this issue so extensively. Some things that I ask here are not kind of due to the fact that I couldn’t figure out thing somehow by myself and the literature only, but because of the feeling to be a bit isolated with my interest in higher category theory here in Karlsruhe and some accumulated needs to discuss this topics with someone to get a better understanding of the subject. So I and my advisor express our thanks for you having an eye on this.
I have just to look again at the identification process for this to make sense.
local isomorphisms are not to be identified with local homeomorphisms
I light of Observation 2.3.123 I see now that the question of trying to compare the notions of local isomorphisms and étale morphisms is not relevant.
I’ll be brief and say more later
This holds for me, too, since (I followed some remarks of David concerning classifying toposes and etendues and accordingly) I am not yet done with the proof that the statement of definition 1 above holds in case of higher locally constant sheaves for topological spaces.
(This is not related but do you know if the category with manifolds (or topological spaces) as objects and proper maps as morphisms is an ∞cohesive site?)
Unless we restrict to compact topological spaces this category has no terminal object, so it is not. Even if we forget about noncompact spaces, I do not know if ”proper stacks” and proper maps coincide with stacks over ”$Mfd^\prop$”.
Hi Stephan,
what’s actually your academic employment situation at the moment?
Ieke Moerdijk and I recently won a grant for a PhD position, with the intended subject being something related to higher stacks and/or higher operads. But the candidate that we had in mind turns out to have other plans. So now we are looking for somebody else.
This position would have to be taken within roughly one year from now.
I am just mentioning this, since I’ll be looking around for candidates now.
Stephan wrote parenthetically
I followed some remarks of David concerning classifying toposes and etendues and accordingly
I admit I am not following this discussion closely, but by etendu you mean the notion as given by Lawvere (Variable sets, etendu, and variable structure in topoi, U. Chicago Lecture Notes, 1976), and mentioned at localic topos?
Hi Urs!
Ieke Moerdijk and I recently won a grant for a PhD position, with the intended subject being something related to higher stacks and/or higher operads. But the candidate >that we had in mind turns out to have other plans. So now we are looking for somebody else. This position would have to be taken within roughly one year from now. I am just mentioning this, since I’ll be looking around for candidates now.
That´s wonderful news! I am highly interested.
what’s actually your academic employment situation at the moment?
I am writing my Diplomarbeit (this corresponds more or less to a masters thesis) at the moment (you probably have noticed:) at the Universität Karlsruhe and one year from now is a realistic timespan to complete my studies there and having received my Diplom. So PhD would be the next step for me. Higher stacks and/or operads interest me and these topics are in line with my current activities. So I would be delighted if this came into existence!
concerning classifying toposes and etendues
I do not know that article by Lawvere but I think the notions are related. The statement which was relevant for me in this context was that a Lie groupoid presents an orbifold if and only if its classifying topos is a proper etendue (proper means here that the diagonal map of the topos is a proper map of toposes  but I didn’t yet find out how these maps are defined). David pointed out to me that the etendueproperty assures then that the presenting groupoid is morita equivalent to an étale one. There are (at least) three characterizations of étendues (but only one is ”localic”): The definition of them given in (SGA4) (mentioned here) was no surprise since it says that an étendue is a category of sheaves on an étale groupoid. So accordingly someone who knew it on the nlab redirected etendue to classifying topos.
proper means here that the diagonal map of the topos is a proper map of toposes
This made me realize that the entries proper map of toposes and compact topos were missing. So I have created stubs for them now. I don’t have time for more, but maybe somebody feels like expanding these further a little.
I am writing my Diplomarbeit […] (you probably have noticed:)
Right, I am well aware that you are writing a thesis, but I forgot of what kind exactly. Okay, good. Can you contact me by email? I should say that I can’t guarantee anything at all, but let’s talk about it.
It seems like it might be slightly more consistent to name the page proper geometric morphism?
I made it redirect, since the other use seems more common, in this case.
But I am not dogmatic about this. Feel free to change it.
Hi Urs,
I thought this evening of how to argue one part of your assumedly statement:
Let $Q\to X$ be a local homeomorphism of topological spaces (where one probably has to assume the topological spaces to be compactly generated). Then there are objects and a commutative diagram
$\array{E&\to& Q\\\downarrow&&\downarrow\\B&\to& X}$in the (∞,1)topos of (∞,1)sheaves on $\Top$ such that the horizontal arrows are effective epimorphisms and the left arrow is $\mathbf{\Pi}$closed.

To show this my first idea is now to do it similarly to the 0dimensional case. It is not yet finished but I post it anyway to see if it goes in the right direction:
I use the notation employed here and the canonical inclusion $\Top_\cg\hookrightarrow \Sh_{(\infty,1)}(\Top)$ and suppress the latter distinction in notation.
Let $(\Pi\dashv \LConst\dashv \Gamma\dashv\Co\Disc):H\xrightarrow{\Gamma}\infty\Gprd$ be a cohesive (∞,1)topos of (∞,1)sheaves on $\Top$, let
$(\mathbf{\Pi}\dashv\flat\dashv \sharp):=(\Disc \Pi,\Disc\Gamma,\co\Disc\Gamma):H\to H$be induced by the adjoint quadruple. $\mathbf{\Pi}$ is idempotent since $(\Pi\dashv \Disc)$ is reflective. $\mathbf{\Pi}$ (as a left adjoint) preserves colimits and (by theorem 2.3.125) pullbacks over discrete objects.
Let $Q\to X$ be a local homeomorphism of topological spaces. Then we find a covering of $\{\U_i\to X\}$ by open sets fitting in the diagram
$\array{U_i\times Q&\to&Q\\\downarrow&&\downarrow\\U_i&\hookrightarrow &X}$Taking the coproduct over all $i$ gives the diagram
$\array{\coprod_i U_i\times Q&\to&Q\\\downarrow&&\downarrow\\\coprod_i U_i&\rightarrow &X}$with the horizontal morphisms being effective (even split) epimorphisms. This remains true if we interpret the diagram in $\Sh_\infty(\Top)$ and request the $U_i\to *$ to be effective epimorphisms.
The morphism $\coprod_i U_i\times Q\to \coprod_i U_i$ is $\mathbf{\Pi}$cosed iff
$\coprod_i U_i\times_{\mathbf{\Pi}(\coprod_i U_i)}\mathbf{\Pi}(\coprod_i U_i\times Q)\simeq\coprod_i U_i\times Q$This would apparently be implied if $\mathbf{\Pi}$ commutes with the product and if the unit of the $(\Pi\dashv \Disc)$ adjunction would evaluate to an isomorphism in $Q$  but at the moment I do not see that. If it works it should come out of some property of Piclosedness I didn’ t realize yet. So far my idea.
Hi Stephan,
this is going very much in the direction that I am imagining, yes.
The only point where I think one needs to refine the argument is where you form the diagram
$\array{ U_i \times Q &\to& Q \\ \downarrow && \downarrow \\ U_i &\to& X \,. }$Remember in the 1categorical setup we had on the left instead of $U_i \times Q$ the product $U_i \times \Gamma_X(Q)(U_i)$ where the second factor is the set = discrete space of local sections of $Q \to X$ over $U_i$. I think here one should do the same, only difference that the sections now form not just a set, but an $\infty$groupoid. But if $Q$ is indeed étale over $X$, the $\infty$groupoid will nevertheless be a discrete infinitygroupoid.
You see, as you correctly notice at the end of your message, for what you have written you’d need that $Q$ itself is a discrete object. But what we really want to be saying here is that $Q$ is “discrete over $X$” or “discrete relative $X$”, if you wish.
In other words, $X$ itself is not in general discrete, and so the total space of a “bundle” $Q \to X$ of discrete $\infty$groupoids or more generally an étale $Q \to X$ will not be discrete either. But if we fix some $U_i \hookrightarrow X$, then the available choices of lifting these through $Q \to X$ will form a discrete $\infty$groupoid, since there will be no continuous choice in the lifts, just discrete choices (and discrete choices of natural transformations between these, and so on).
Hi Stephan,
part of what I just said I have now made explicit as example 2.3.131, remark 2.3.132.
Something wrong with the link in 38.
Hi Urs,
You see, as you correctly notice at the end of your message, for what you have written you’d need that itself is a discrete object.
I see. In my setup  if I had stuck resolutely to my assumption of $Q$ being discrete it would have followed since then the application of $\mathbf{\Pi}$ then does not change $Q$, but since indeed it was my problem to interpret the formula $\Gamma_X(Q)(U_i)$ correctly I didn’ t. Thanks for explaining the correct notion of ”relative discreteness”!
Something wrong with the link in 38.
Thanks, I have fixed it.
Since a while ago, after some of these socalled updates, Firefox likes to strip off the “http://” when I copy and paste links. Sometimes, sometimes not.
if I had stuck resolutely to my assumption of Q being discrete it would have followed
Yes!
Hi Urs,
part of what I just said I have now made explicit as example 2.3.131, remark 2.3.132.
Fine. This completes then this direction of the proof and it remains just the other implication to have the basic result of the characterization of étale morphisms via locally constant (∞,1)sheaves settled.
(There are two typographical inaccuracies which ocurred to me reading example 2.3.131, remark 2.3.132: In the proof of prop. 2.3.128 the pullback in the diagram should be $c_\Pi f=Y\times_{\Pi (Y)}\Pi(X)$ instead of $X\times_{\Pi (Y)}\Pi(X)$. In the diagram of remark 2.3.132 there is a bracket shifted: this should be $\Disc(A)//Aut(A)$ instead of $\Disc(A//Aut(A))$, I guess)
This completes then this direction of the proof and it remains just the other implication to have the basic result of the characterization of étale morphisms via locally constant (∞,1)sheaves settled.
The proof of which statement do you have in mind now?
I was thinking that for étale $\infty$stacks what we give is a definition (étale = covered by locally constant). Nothing to be proven at that point. But then next one wants to see that this reduces to what it should reduce to in low degree.
I think we checked this in lowest degree. And to my mind also in the next degree, see the discussion above. So that looks good.
The next steps, to my mind, have to be providing more context.
The main next step is: show that étale simplicial manifolds are presentations for étale $\infty$stacks. What should this actually mean for us? It should mean something like that “the source map” (or the target map) from the homsimplicial manifold to the base manifold is an étale morphism of $\infty$stacks. This statement may need a bit of adjusting…
In the proof of prop. 2.3.128 the pullback in the diagram should be $c_\Pi f=Y\times_{\Pi (Y)}\Pi(X)$ instead of $X\times_{\Pi (Y)}\Pi(X)$.
Right, thanks! I have fixed that now. Thanks.
In the diagram of remark 2.3.132 there is a bracket shifted: this should be $\Disc(A)//Aut(A)$ instead of $\Disc(A//Aut(A))$, I guess)
No, that’s actually correct as stated. $A$ is a bare $\infty$groupoid, and $Aut(A)$ is a bare infinitygroupoi. $A//Aut(A)$ is the corresponding bare action groupoid. The functor $Disc$ sends all this to smooth $\infty$stacks, by regarding bare $\infty$groupoids as discretely smooth $\infty$groupoids.
On the other hand what one could do is use
$Disc(A // Aut(A)) \simeq Disc(A) // Disc(Aut(A))$(two Discs on the right!). Because $Disc$ is a left adjoint, and so preserves the weak quotient of $A$ by $Aut(A)$.
You know, Stephan, I am actually suffering from a confusion that we have been carrying on all along:
maybe it is not really clear to me what the “étale 2morphism” (or “local 2homemorphism”, if you wish) is that we want to be associating with an étale groupoid.
I think the “other direction” that Stephan wants to show is that in the case of topological spaces, your definition of etale collapses to the known one. There are various properties that your notion of etale morphism should satisfy for it to be “any good” in the sense that the machinery of “ordinary” topological etale stacks goes through. The most important is the full subcategory of the slice category over an object consisting of etale morphisms needs to be a topos (or in the higher setting, an infinity topos). Here’s something that might be useful: let $G$ be any full subcategory of $Sh_J(C)/X$ for $X$ a sheaf. Notice that $$Sh_J\left(C,\right)/X \cong Sh_{J'}\left(C/X\right),$$for$J’$the induced Grothendieck topology. Restrict$J’$to$G$. Let$$f:G \to Sh_J(C)/X$ be the full and faithful inclusion. Then $$F_!:Sh_{J'}\left(G\right) \to Sh_J(C)/X$$ is full and faithful (so its image is a topos).
I would like to mention another thing though: Etale stacks are not god given! They depend on your choice of site. This is because the etale topological stack which are actually sheaves are precisely topological spaces (i.e. representables), or in the differentiable setting manifolds (which may or may not be representable depending on your choice of site). So basically, you want the only $1$sheaves which are etale stacks to be locally isomorphic to representables. This should somehow be worked in. This is “evil” of course, since you can also equip your topos with the canonical topology so that everything is representable (hence etale stacks cannot be totally intrinsically defined). Basically, what I’m saying is you need not just a topos, but some sort of “geometry” around which GIVES RISE to your topos and also to a natural definition of etale stack.
Combining these two remarks, the first is only useful, if you can show the image topos actually lands in maps into $X$ from a locally representable sheaf, when $X$ itself is locally representable. If this is not clear, I can explain further.
I am actually suffering from a confusion that we have been carrying on all along
I am really sorry for this! After we clarified that we do not speak of local isomorphisms instead of local homeomorphisms I have not been confused any more, I claim. But I will be more extensively in remarking what actually I mean.
I was thinking that for étale $\infty$stacks what we give is a definition (étale = covered by locally constant).
I think the “other direction” that Stephan wants to show is that in the case of topological spaces, your definition of etale collapses to the known one.
There are various properties that your notion of etale morphism should satisfy.
Of course it is a definition and by ”the other implication” I did not mean to indicate that there is a fxed notion of ”higher étale morphism” to check. But as David says there are properties which the morphism which is covered by our definition should satisfy. Maybe we can talk about this set of properties we shoud require. For instance there are formally etale morphisms (with respect to some infinitesimal (cohesive) neighbourhood) and Davids remarks to consider and I didn’t yet think about the definition / statement with local diffeomorphisms instead of local homeomorphisms. So at least for me there are yet details to clarify in the definition / statement.
Hi Stephan,
ah, there was a misunderstanding. I didn’t mean that you were the source of any confusion of mine!
I was just expressing a puzzlement of mine about a general point related to our discussion.
But I got over my puzzlement. (If it wasn’t clear what I was puzzled about, maybe it’s best not to rehash it, now that it’s over ;)
$\,$
Here is another question that I was thinking of: how far do we get if we consider the following notion:
morphisms $Y \to X$ such that all spaces of local sections are discrete.
Here is what I mean in more detail. Let $p : Y \to X$ be any morphism in any topos (or higher topos) $\mathbf{H}$. Then for $\phi : U \to X$ any morphism, the set (or $\infty$groupoid) of sections of $p$ over $U$ is the pullback (or $\infty$pullback) $\Gamma_U(Y)$ in
$\array{ \Gamma_U(Y) &\to& \mathbf{H}(U, Y) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{H}(U,p)}} \\ * &\stackrel{\phi}{\to}& \mathbf{H}(U, X) } \,,$where $\mathbf{H}(,)$ is the homset (or hom $\infty$groupoid).
We can here replace the hom with the internal hom $[,]$ of the topos. This leads us to the internal object of local sections $\bar \Gamma_U(Y)$, which is the pullback
$\array{ \bar \Gamma_U(Y) &\to& [U, Y] \\ \downarrow && \downarrow^{\mathrlap{[U,p]}} \\ * &\stackrel{\phi}{\to}& [U, X] } \,,$now taken as a pullback (or $\infty$pullback) in $\mathbf{H}$.
If $\mathbf{H}$ is cohesive (and actually more generally, but let’s think of cohesive), then we may ask if the object of sections $\bar \Gamma_U(X)$ is a discrete object.
I am wondering if this might be a better way to go about characterizing étale morphisms $Y \to X$ than by the “covered by covering maps”strategy that we looked at before. But I guess there may be some extra condition one may have to impose. Hm..
So I guess I am trying to get at the following:
For $p : Y \to X$ a morphism in our topos (or $\infty$topos) $\mathbf{H}$ we obtain a presheaf (or $\infty$presheaf) on the “slice site” $\mathbf{H}/X$ (the slice topos regarded as a site by equipping it with the canonical topology)
$\Gamma_{}(Y) : (\mathbf{H}/X)^{op} \to \mathbf{B}$(where $\mathbf{B} = Set$ or $\mathbf{B} = \infty Grpd$)
by sending any $U \to X$ to the set (or $\infty$groupoid) $\Gamma_U(Y)$ of local sections, as in my above message.
But we also get an $\mathbf{H}$valued presheaf
$\bar \Gamma_{}(Y) : (\mathbf{H}/X)^{op} \to \mathbf{H}$by taking instead the objects of internal sections, also as in the above message.
(These presheaves will indeed be sheaves, but let’s not worry about that for the moment.)
So then we can ask wether $\bar \Gamma_{}(Y)$ happens to factor through discrete objects as
$\bar \Gamma_{}(Y) : (\mathbf{H}/X)^{op} \to \mathbf{B} \stackrel{Disc}{\hookrightarrow} \mathbf{H} \,,$where $Disc : \mathbf{B} \hookrightarrow \mathbf{H}$ is the inclusion of discrete objects into $\mathbf{H}$.
What I am getting at is that we might want to say that $p : Y \to X$ is étale, if the sheaf (or $\infty$sheaf) of internal sections factors through discrete objects this way.
Hi Urs,
I didn’t mean that you were the source of any confusion of mine!
I’m quite relieved to hear this!:))
…then we may ask if the object of sections is a discrete object.
The form of this problem somehow reminds me of some statement: If $X\to Y$ is a morphism in ∞Grpd and $X$ is $\kappa$small then $Y$ is $\kappa$small iff all its fibers $Y\times_{\{x\}}$ is $\kappa$small where $x\in X$ are objects (if I remember that correctly). I have just to check if this is related at all (in particular I know that this intuition comes from the well known common misconception ”discrete=finite”).
What I am getting at is that we might want to say that $p : Y \to X$ is étale, if the sheaf (or $\infty$sheaf) of internal sections factors through discrete objects >this way.
Yes, I understand what you mean. As a slogan it is clear that ”locally constant objects are discrete”  so, maybe expressing our former attempt this way has some technical advantages. I have to think about. Hopefully tomorrow I come up with some some more detailed remarks.
What do you think about the role the notion of formally étale morphism can play here?
Basically, what I’m saying is you need not just a topos, but some sort of “geometry” around which GIVES RISE to your topos and also to a natural definition of etale stack.
Hi David, I guess you refer to a ”geometry” as in Jacob Luries book on higher algebra?
The form of this problem somehow reminds me of some statement:
The statement that you have in mind is the definition of “relatively $\kappa$compact objects” as stated here in the entry object classifier in an (infinity,1)topos.
I wouldn’t say that this is too closely related to what we are discussing. But of course there is a common theme, in that in both cases we are characterizing fibers of a morphism, that’s true.
the well known common misconception ”discrete=finite”).
Well there is also a common good conception here, which is “finite implies discrete”, which is true for geometric structures like manifold structure, etc.).
What do you think about the role the notion of formally étale morphism can play here?
You have a good point there.
I am feeling a little stupid for not long ago having made the relation to the notion of formal étaleness in a cohesive $\infty$topos as in section 2.4.1.3.
I should think about that…
Just a bit more rambling about this idea of characterizing “discretely sectioned objects”.
For $\mathbf{H}$ any topos and $X \in \mathbf{H}$ any object, we get the corresponding etale geometric morphism
$\mathbf{H}_{/X} \stackrel{\leftarrow}{\to} \mathbf{H}$which exhibits the slice topos as a topos over the base topos $\mathbf{H}$.
So then we can consider the $\mathbf{H}$valued hom
$[,]_X : (\mathbf{H}_{/X})^{op} \times \mathbf{H}_{/X} \to \mathbf{H} \,.$In terms of this every object $(P \to X) \in \mathbf{H}_{/X}$ represents an $\mathbf{H}$valued functor
$y(P) := [, P]_X : (\mathbf{H}_{/X})^{op} \to \mathbf{H}$This is to be thought of as assigning to each object $(U \to X) \in \mathbf{H}_{/X}$ the internal object of sections of $P \to X$ over $U$.
By the suitable over$\mathbf{H}$ analog of the standard story it should be true that every canonical $\mathbf{H}$sheaf on $\mathbf{H}_{/X}$ is of this form, so that we identify objects of $\mathbf{H}_{/X}$ with $\mathbf{H}$sheaves on the canonical site of $\mathbf{H}_{/X}$.
(I always need to beef up my over$\mathbf{H}$toposophy, so take this with a grain of salt for the moment. I am just trying to sketch some ideas for how to proceed here.)
But now assume that $\mathbf{H}$ is cohesive. Then we can consider the full subcategory $DiscFib_{X} \hookrightarrow \mathbf{H}_{/X}$ of $\mathbf{H}_{/X}$ on those objects $P$ for which $y(P) = [, P]_X$ factors through the discrete objects $Disc : \mathbf{B} \hookrightarrow \mathbf{H}$.
I guess with the above supposed identification of $\mathbf{H}_{/X}$ with its canonical $\mathbf{H}$sheaves and using that $Disc : \mathbf{B} \hookrightarrow \mathbf{H}$ is a reflective subcategory whose reflector preserves products, it should follow rather immediately that also $DiscFib_{X} \hookrightarrow \mathbf{H}_{/X}$ is a reflective subcategory whose reflector preserves finite products.
But since there is no reason so far that the reflector preserves all finite limits, $DiscFib_X$ has no reason to be a topos. This cannot yet quite be the category of étale objects over $X$ that we are after.
I think a quick example shows that what is missing is some openness condition.
For let $X$ be any topological space and $x : * \to X$ the inclusion of any one point. Take this to be our $P \to X$. Then the internal space of sections of $* \to X$ over any continuous map $U \to X$ is discrete, because it is the singleton space if $U \to X$ factors through $x : * \to X$, and empty otherwise. But $x : * \to X$ is of course not étale, because the point $*$ is closed and not “spread out” a little over $X$.
So within the above subcategory $DiscFib_X$ one might want to look for a further reflective subcategory such that the total reflector from $\mathbf{H}_{/X}$ then preserves all finite limits, and that might then be the subtopos
$Etale_X \hookrightarrow DiscFib_X \hookrightarrow \mathbf{H}_{/X}$that we are after.
Something like this.
P.S. I have to interrupt this discussion now, since I need to be looking into some exceptional geometry of supergravity as here, for the moment…)
P.P.S For emphasis I should say that this discussion here is not really aimed at the notion of (higher) étale groupoids, despite the title of the thread. But at the notion of étale maps. The two are related, but that relation deserves a separate discussion.
Hi Urs,
based on you recent thoughts and the vague idea that the identity $\Et /X\simeq \Sh(X)$ which holds for topological spaces should become in higher dimension rather an inclusion ”$\Et_{(\infty,1)}/X\hookrightarrow \Sh_{(\infty,1)}(X)''$ the reflective embedding with limit preserving left adjoint
$\array{Etale_X&\stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}&H/X\\\searrow&&\nearrow\\&\Disc\Fib_X &}$aka exact localization factoring over $\Disc\Fib_X$ looks like ”sheafifying once more” by localizing at morphisms of $\Disc\Fib_X$.
In this form we could then try to apply the recognition lemma for exact localizations:
Let $S_0:=(\Disc\Fib_X)_1$ denote the class of morphisms in $\Disc\Fib_X$, let $S$ denote the smallest strongly saturated class of morphisms containing $S$.
(A class $S$ of morphisms is strongly saturated if it is (1) pushout stable, (2) the restriction to $S$ of the arrow category $\Arr(C)_S$ has all (∞,1)colimits and (3) it satisfies 2outof3.)
Then the above is an exact localization if pullbacks of $S_0$ land in $S$.
$\,$I don’t know yet if this reformulation has any advantages or works at all but I will look at it tomorrow.
ps. I am not sure what exactly the compiler dislikes about the above diagram.
Hi Stephan,
it’s true that we could alternatively try to identify the morphisms at which the desired inclusion $Etale_X \hookrightarrow \mathbf{H}/X$ is the corresponding localization.
But then the question is: which set of morphisms should that be? I don’t seem to see a natural candidate right now. This need not mean that there is not one. But notice that the suggestion
Let $S_0 = DiscFib_X$
cannot be the answer. The morphisms in $S_0$ are going to be inverted by the localization. So localizing at this $S_0$ will make all maps between discrete fibrations into equivalences. Since the identity map $X \to X$ will be a discrete fibration over $X$, and since this is the terminal object in $\mathbf{H}/X$ and in $DiscFib_X$, this would collapse everything in $DiscFib_X$ to a point.
The idea with finding the set of morphisms at which a given subcategory is a localization is somewhat opposite to this: a morphism is going to be in $S$ if it looks like a weak equivalence when homming it into an object of the subcategory.
You can see this from the reflection adjunction $(Reflector \dashv inclusion)$. If $P$ is in the subcategory and $f : A \to B$ is in $S$ (hence inverted by the reflector) , then by the adjunction
$\mathbf{H}_{/X}(A \to B, P) \simeq Etale_X(Reflector(A \to B), P)$and so $\mathbf{H}_{/X}(P, B) \to \mathbf{H}_{/X}(P, A)$ is an equivalence.
So the question to ask is: can we identifiy a good set $S_0$ of morphisms over $X$ such that homming them into an étale map over $X$ makes them an equivalence?
Hi Urs,
the suggestion $S_0 = DiscFib_X$ cannot be the answer
Ok, this was a mistake. The localization process in general and in particular that this choice would contract $\Disc\Fib_X$ was clear to me  but what to outcome of a correct choice should be was not  and still is not. In particular with regard to the additional condition making the localization exact. Without such intuition which you say you lack, too, at the moment I guess it is better to look at the openness condition you mentioned that might give an exact localization. I will try with class of open maps in a topos.
ps. The last days I was somehow afflicted with lack of time but off tomorrow it will be better.
Hi Urs,
I think a quick example shows that what is missing is some openness condition.
yes, I think this is true:) In the (very enlightening) paper (Ieke Moerdijk and Andre Joyal: A completeness theorem for open maps doi ) are given axiomatizations of the notions of (classes of) open and of étale maps in toposes (and under additional asuumptions in any category). These axioms are (given in open map but since the collection axiom needed for the completeness theorem below lacks there, I insert them here for convenience):
Let $\array{Z&\to Y\to^p&X\\\downarrow^g&&\downarrow^f\\B&\to^h}&H$ be a topos.
A class $R$ of $H$morphisms is called class of open maps if (A1)(A6) are satisfied, $R$ is called class of étale maps if (A1)(A5) and (A7)(A8) are satisfied. The axiom (A9) is called collection axiom.
(A1) any isomorphism belongs to $R$ and $R$ is closed under composition.
(A2)(stability) arbitrary pullbacks of a morphism in $R$ is in $R$
(A3)(descent) if in the pullback square in $H$ the left arrow is in $R$ and the bottom arrow is epi then the right arrow is in $R$
$\array{X&\to&Y\\\downarrow&&\downarrow\\X^'&\to& Y^'}$(A4) for any set $I$ the morphism $\coprod_{i\in I}1\to 1$ is in $R$
(A5) for any family of arrows $(Y_i\to X_i)_{i\in I}$ the morphism $\coprod_{i\in I}Y_i\to \coprod_{i\in I}X_i$
(A6)(quotient) for any factorization $f\circ p=q$ such that $p$ is dpi and $g$ is in $R$ then $f$ is in $R$.
$\array{ Y &&\stackrel{p}{\to}&& X \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}} \\ && B }$(A7) if $f:Y\to X$ is in $R$ then its diagonal $\Delta_f:Y\to Y\times_X Y$ is in $R$
(A8) if in the pullback square the left arrow and the bottom arrow are in $R$ and the bottom arrow is epi then the right arrow is in $R$.
(A9)(collection) if $p$ is epi and $f$ is in $R$ there exists a quasi pullback with $h$ epi and $g$ is in $R$.
The collection axiom appears in
Let $R$ be a class of open maps satisfying the collection axiom (A9) in a topos $H$.
There is a topos $T$ and a geometric morphism $F:\Arr(T)\to H$ such that $R$ is induced by the canonical class of open maps in $\Arr(T)$;
i.e. $f\in H_1$ is open iff $F^* f$ is a quasi pullback square in $T$.
and
Let $H$ be a topoos satisfying the collection axiom (A9), let $X\in H$ be an object.
Then there is a coreflective geometric morphism $(i^*\dashv i_*):H/X\to\Et/X$
Apparently one can write down these axioms in higher dimension, too. So I guess my worthwhile agenda is now to consider how exactly they categorify, which statements of this paper hold in higher dimension, and see how they simplify if we assume the (higher) topos to be cohesive (and accordingly what this has to do with discreteness). What do you mean? Did you know this paper?
One thing to consider is how the open maps in $Top$ are generated by the class of maps $\coprod U_i \to X$, the archetypal open maps. Quite possibly we can generate, given the other axioms, from the inclusions $U_i \hookrightarrow X$. There are analogies with Zariski maps of rings and the etale topology on $Sch$.
Recall also definition 2.3 (local homeomorphism) here: http://ens.math.univmontp2.fr/~toen/cours1.pdf (Toen’s Master course)
As for ’geometries’, see section 1 here: http://ens.math.univmontp2.fr/~toen/cours2.pdf
Did you know this paper?
Yes. I had added that material to open map in revision 4.
Admittedly, though, I hadn’t thought of it in the present discussion.
What do you [think]?
Yes, I think this is well worth looking into. I’ll try to get back to you later. Need to be dealing with some other things right now.
you
Thanks David and Urs  of course anyone is addressed and invited to contribute to this discussion; I addressed to Urs in first place just because we started this discussion.
Recall also definition 2.3 (local homeomorphism) here: http://ens.math.univmontp2.fr/~toen/cours1.pdf (Toen’s Master course)
The definition of open maps given here is less general than the one given by the (A1)(A9).
As for ’geometries’, see section 1 here: http://ens.math.univmontp2.fr/~toen/cours2.pdf
Here is the given the characterization of open maps via the ”archetypical open maps” you mentioned It is interesting here that he uses representable presheaves for his definition of open maps since above we tried it with the other inclusion of the base site in its sheaf category: locally constant sheaves.
There are analogies with Zariski maps of rings and the etale topology on Sch.
Yes, I have to look at that more closely  in principle this is a special case of that of localic presheaves…
The definition of open maps given here is less general than the one given by the (A1)(A9).
ah, ok. I didn’t compare. Also I only skimmed this discussion, so may not have followed all your details.
Thanks David and Urs
Well, now we have to disambiguate: me and David C :)
now we have to disambiguate
At least english grammar in many cases doesn’t force me to distinguish between singular and plural when I write ”you” :)
I only skimmed …
…Toens masters course since I guess the material presented there is covered and further developed by Luries Structured Spaces V.
There are analogies with Zariski maps of rings and the etale topology on Sch.
To return to your remark on the étale topology: I begin to get a more complete idea of the topic now. The étale site defined by the étale topology is a pregeometry and its enveloping geometry is one motivating example for a geometry (for structured (infinity,1)toposes) David (:) mentioned above. This notion reduces to what Toen and Ieke Moedijk do in giving some distinguished class of ”prototypical” open resp. étale maps.
But we meant to find an ”intrinsic” description of open and étale maps and and mentioning an underlying geometry is not intrinsic as I understand this term.
But we meant to find an ”intrinsic” description of open and étale maps and and mentioning an underlying geometry is not intrinsic as I understand this term.
Right, that was exactly the idea. That in a cohesive (higher) topos there should be a canonical notion of étale map, because there is already a canonical notion of covering étale map (an étale map whose fiber is the same everywhere).
That may or may not turn out to be true in the end. But even if it should turn out not to be true, it would still be interesting to see to which extent it is true. For instance the $\infty$category of “discretely fibered” maps that I mentioned above may already have its use.
From another persective, one could ask: does a cohesive higher topos induce its own canonical geometry (in the sense you mentioned)?
But if these questions start be more of a hinderance than an inspiration, one should start to try other approaches.
does a cohesive higher topos induce its own canonical geometry
this I think is the key question. Why? Because the toy example Urs started with used covering spaces, which trivialise over, you guessed it, open covers. Just saying all the fibres are isomorphic seems weaker to me (but I’m happy to be proved wrong). But then how do we define fibres of local isomorphisms for a general higher topos? I might have missed this somewhere in the flurry of discussion above.
But then how do we define fibres of local isomorphisms for a general higher topos?
First a nitpicky remark, repeating comments that I made before:
Where you say “local isomorphism” I do hope you mean “local homeomorphism”. Despite the similar terminology, these notions have nothing to do with each other! They are sort of dual to each other: a local isomorphism is, for typical situations that you may have in mind and roughly: something that becomes an iso in the vicinity of points in the codomain. A local homeomorphism becomes an iso in the vicinity of points in the domain.
So then let’s look at this question:
But then how do we define fibres of local homeomorphism for a general higher topos?
This has an obvious answer: we look at homotopy fibers. :)
does a cohesive higher topos induce its own canonical geometry
I am looking for this now. In the above discussion I see three candidates for an admissible structure on a (∞,1)category $G$ which hopefully lead to a canonical geometry $G^'$ such that $O:G^'\to H$ is left exact and ”preserves admissible covers” and hence gives a $G$ structure on a cohesive (∞,1)topos $H$  namely: $G^\ad_0:=H_0$ and
$G^\ad_1:=$ morphisms effectively covered by Piclosed morphism, or
$G^\ad_1:=\Etale_X\hookrightarrow \Disc\Fib_X \hookrightarrow H/X$ where the reflector of the composite inclusion preserves finite limits, or
$G^\ad_1:=$ formally étale morphisms
As it stands none of these can be a geometry since $H$ is not essentially small. But this may be fixed by sorting out or by starting with a $G$ which is.
Formally étale morphisms satisfy 2outof3, are stable under retracts and are stable under pullback if the full and faithful embedding $u^*:H\to H_\th$ into a cohesive neighbourhood of $H$ preserves pullbacks so in case 3 we have an admissible structure.
I don’t have the head free for this right now, but you have a very good point by focusing attention back to the formall étale maps. We should push that, since here we really know a good deal of what’s going on.
Formally étale morphisms satisfy 2outof3 and are stable under pullback if the full and faithful embedding into a cohesive neighbourhood of $\mathbf{H}$ preserves pullbacks.
Right. Hm, do we have this for our preferred models? I’d need to think about that. Right, that would be good…
In fact, according to this proposition, the cohesively formally étale morphisms between smooth manifolds are ordinary étale maps.
Right, very good point, I should have made that connection earlier. It’s pretty compelling, now that I think about it.
Also, it makes good sense in view of the notion of “geometry”: the extra information of “admissibility” is somehow encoded in the extra choice of infinitesimal cohesion. Makes perfect sense, now that you made me think this way.
A necessary condition for 1. to be admissible is that $\Pi$closed morphisms are themselves closed under pullback and I guess Piclosed morphisms are closed under pullback:
If $f:X\to Y$ is Piclosed and $f^':X^'\to Y^'$ is a pullback of $f$ there is a pasted an (∞,1)pullback diagram
$\array{X^'&\to&X&\to& \Pi X\\\downarrow&&\downarrow&&\downarrow\\Y^'&\to&Y&\to &\Pi Y}$By naturality and universality of the $(\Pi\dashv\Disc)$unit this diagram is equivalent (I have to check this again tomorrow) to
$\array{X^'&\to&\Pi X^'&\to& \Pi X\\\downarrow&&\downarrow&&\downarrow\\Y^'&\to&\Pi Y^'&\to &\Pi Y}$where the left pullback shows that $f^'$ is $\Pi$closed.
I don’t have the head free for this right now
I just posted it here as a kind of status report on what I’m currently thinking of, so there is no need that you comment it instantly.
A necessary condition for 1. to be admissible is that $\Pi$closed morphisms are themselves closed under pullback and I guess Piclosed morphisms are closed under pullback:
Yes, indeed. They are the right class in a factorization system.
I just posted it here as a kind of status report on what I’m currently thinking of, so there is no need that you comment it instantly.
Sure, and thanks for this. But I did want to comment instantly, because you made a good comment! So I just wanted to warn that even though I am commenting, my comment is not to be read as the result of intesive thinking. ;)
Formally étale morphisms satisfy 2outof3
I must correct this. If we consider the triple $\{f,g,f\circ g\}$ the implication ”$f$ and $g\circ f$ formally étale entails $g$ is formally étale” is not shown. Nevertheless formally étale maps satisfy the admissibility axioms.
There was also a typo in proposition 3 in formally etale morphism which I corrected
Yes, indeed. They are the right class in a factorization system.
That’s fine! Then they give an admissible structure, too. In fact the right class of every orthogonal factorization system gives an admissible structure.
 $G^\ad_1:=$ morphisms effectively covered by Piclosed morphism
Yes, indeed. They are the right class in a factorization system.
I am trying to figure out if factorization systems are the right notion to describe ”étale morphisms”: The $(\Pi_\eq\dashv\Pi_\cl)$ orthogonal factorization system is dual to the factorization system $(\Pi_cl^\op,\Pi_\eq^\op)$ on $H^\op$. Since we are interested in diagrams of type
$\array{A&\to&X\\\downarrow&&\downarrow\\B&\to&Y}$with the left arrow (a reversed) $\Pi$closed and the horizontal morphisms being effective epimorphisms, we could consider ”$E$orthogonal factorization systems” where the filling diagonal is required only for squares where the horizontal morphisms are in the class $E$ of effective epimorphisms. This idea occurred to me since in the motivating example of a coverage (by a covering space) we had indeed a unique diagonal.
Another question now, that we have two (different) canonical admissible structures on a cohesive topos  is, what they have to do with each other: I tried to compare them via the factorization of the global section geometric morphism of the infinitesimal cohesive neighbourhood of $H$ through $H$.
I have added more standard references to étale groupoid.
Added to the Definitionsection at étale groupoid some paragraphs on the Morita/stackyinvariant formulation via “foliation groupoids” and the characterization theorem by CrainicMoerdijk.
Added a brief remark on “noncommutative Gelfand duality for etale Lie groupoids”.
I have added to etale groupoid some of the relevant facts proven in Carchedi 12.
His result which you quote is quite similar to main results of a series of important works of Pedro Resende on relations between etale groupoids and inverse semigroups.
Of course, thank you for calling our attention to Carchedi’s paper ! Pedro was mainly working in localic context and the 2categorical and Morita aspects were studied more recently than his main known results on the subject. He has also some very nice works in preparation.
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