I think we agree on this, I understand this, just the adjunction between simplicial nerve does not need to know of choices in weak homotopy classes unless you prove something more specific. But never mind, we can talk about that some time in future.

]]>Take X to be the point, for a moment. Then the equivalence becomes that between topologivcal space and simplicial sets. The statement is that the geometric realization of a simplicial set that is just a set is a discrete topological space. An etale space over the point. The weak equivalences between such etale spaces over the point are just isomorphisms of sets.

Do we agree on this? Since the general situation is in the overcategory, it is an X-parameterized version of this situation.

]]>An etale space is like a set in Top/X: it is an object in there that is fiberwise a set

It is a good intuition but no refinements of that intuition are supplied by Lurie's theorem.

]]>You see, you are trying to talk about various PARTS of left and right hand side of adjunction at infinity level, as if we had faithful inclusions (but everything is just up to homotopy) and we do not know that the natural functors (which are not 1-categorically faithful) are compatible with the adjunction. Simplicial nerve can not even SEE the details which are beyond simplicial models, nor a quillen equivalence can. The fact that you need assumptions like paracompactness means you do preparation for doing just weak homotopy information.

No, I do not agree that zig zag starting with an image of etale space in simplicial nerve and ending in some other object in the simplicial nerve is an equivalence of etale spaces. And even if it is, it does not describe the infintge catgeorical analogues in topological terms, but in terms of simplicial localization.

]]>So you have a zig-zag of isomorphisms of etale spaces then. That's still an isomorphism of etale spaces.

The first one is etale space, the other spaces in zig zag are not necessarily etale, nor we can guarantee that the last one is if the first one is without stating more preciuse form of a theorem.

]]>but Lurie's theorem is not about Top/X but about its simplicial nerve. So zig zags are also allowed in the story;

So you have a zig-zag of isomorphisms of etale spaces then. That's still an isomorphism of etale spaces.

In fact his theorem does not say anything about etale spaces, because the category of etale spaces is not at either side of his adjunction.

Well it is the subcategory in Top/X that is in the image of the oo-sheaves that are set-valued.

And what the theorem does on top of this is to say what the higher analogs of etale spaces are: namely the image in Top/X of n-groupoid valued sheaves in Sh_oo(X).

Just we know indirectly that WEAK homotopy types of etale spaces should correspond to a sub-infty,1-category at the left side.

I am not sure what else I can say in reply to this. Do we agree that an zig-zag of weak equivbalences between etale spaces in Top/X is just an iso of etale spaces? Because each weak equivalence in the zig-zag is. This is like saying that a weak equivalence between two sets cannot be anything but an isomorphism of sets. A zig-zag of weak equivalences of sets is still just an isomorphism of sets. An etale space is like a set in Top/X: it is an object in there that is fiberwise a set.

]]>The paracompactness is a minor point and my complaints were more serious.

Okay. You know, morphisms of etale spaces are usually defined to be the morphisms in Top/X between etale spaces.

Yes, but Lurie's theorem is not about Top/X but about its simplicial nerve. So zig zags are also allowed in the story; plus Lurie's theorem does not say (as I wrote about several times already) that doing the adjunction back and forth can be strictified in such a way that starting with etale space you get again an etale space. In fact his theorem does not say anything about etale spaces, because the category of etale spaces is not at either side of his adjunction. Just we know indirectly that WEAK homotopy types of etale spaces should correspond to a sub-infty,1-category at the left side.

]]>Okay, no, that's good, don't worry. Sometimes one just needs to sync into each other's frequency, even if the item under discussion is clear to everyone internally.

]]>I got the wrong end of the stick by skimming over the profusion of comments that came up during yesterday. I think I read this (by you, Urs)

weak equivalences in Top/X are stalk-wise weak equivalences.

and got confused. I'm wasn't sure what you mean by a stalk-wise weak equivalence for a map in Top/X. If you mean fibre homotopy equivalence then I agree. I just thought there was some tricky passing to stacks or something that was implicit - I'm probably just confusing myself.

morphisms of etale spaces are usually defined to be...

yeah, I know (wry grin). I haven't lost *all* my marbles.

Okay. You know, morphisms of etale spaces are usually *defined* to be the morphisms in Top/X between etale spaces. Etale spaces are defined as a full subcategory on Top/X on which a certain adjunction produces an equivalences. Isn't that actually the starting point of all of the discussion here?

I'm probably mixed up too.

What's an example of a morphism in Top/X between two Etale spaces that's not a morphism of Etale spaces?

welll... they are are the same thing. I'm probably not quite familiar with the sheaf/stack side of things, and I was taking the statement about fibrewise iso/equivalence to mean that fibre by fibre there is an iso/equivalence (not induced by a morphism of etale spaces). This is clearly false, but if you didn't mean that, then I apologise for being dense.

For the second point I was going from Zoran's comment

Take ANY topological space X, not necessarily paracompact Hausdorff, and then take appropriate weak n-category ... internal to Top/X

and I agree that the assumption of paracompactness is for homotopy theory too strong. To me this says that we should be looking for another theorem going beyond Lurie's, if this is possible.

edit: I was going to discuss Lurie's technical requirements for paracompactness, but this is better to be done on the page itself.

]]>but a fibrewise isomorphism does not imply an isomorphism of etale spaces.

Okay, I am probably mixed up, then. What's an example of a morphism in Top/X between two Etale spaces that's not a morphism of Etale spaces?

but if we are interested in more than just paracompact spaces?

The theorem byy Lurie that we are talking about uses the assumption of paracompact base space. Without that, we don't have that theorem.

]]>Zoran wrote:

Edit (clarification): In fact Top/X should be replaced by n-category of internal (n-1)-categories in usual 1-categorical Top/X on the left, or a variant thereof, if I wasn't clear enough. Etale 2-space should be a special object there for n=2.

As in my thesis :)

Urs wrote:

If the morphism is an equivalence, this must be an isomorphism for each fiber, evidently, because in Top/X everything must work fiberwise. So equivalences of etale spaces regarded as objects in the oo-category Top/X are just ordinary isos of etale spaces.

but a fibrewise isomorphism does not imply an isomorphism of etale spaces.

a weak equivalence of simplicial presheaves on a paracompact topological space is one that is stalkwise a weak equivalence of simplicial sets.

but if we are interested in more than just paracompact spaces?

]]>Let's do the eale spaces first. I have trouble seeing what flexibility you seem to see.

we have two etale spaces E1 and E2 over X regarded as objects in Top/X. Any morphism between them is a morphism of spaces over X. That means it has to be a map of the total spaces such that for each point x in X the fiber E1_x is mapped to the fiber E2_x .

If the morphism is an equivalence, this must be an isomorphism for each fiber, evidently, because in Top/X everything must work fiberwise. So equivalences of etale spaces regarded as objects in the oo-category Top/X are just ordinary isos of etale spaces.

Or what am I missing?

]]>If you have a category with pullbacks, then internal categories correspond to presheaves of categories; and all possible categorifications and strengthenings of it (but here sizes may matter). I thought that this could be combined with the usual 1-categorical statement on etale spaces to some benefit.

Look at it on the right side of the equivalence:

The right hand side is simplicial, I mean no localization is done, just taking simplicial presheaves. The left hand side is the one which has information on potentially bad local topology. Any space over X. Now in a weak homotopy class which is sense at the level of simplicial localization the class of an etale space contains also many representatives which are very different in the sense not felt by weak homotopy type and by simplicial model. These ones worry me. Lurie is having a statement about weak homotopy type not about genuine Top/X.

So weak equivalences between oo-stacks that happen to be ordinary sheaves are ordinary isos of sheaves.

What guarantees that if you start with an etale space take its image in simplicial nerve of Top/X do Lurie adjunction and return back to the simplicial nerve, that you will get back a weak homotopy class of an etale space ? Even more for internal n-categories in etale spaces for higher n. This would need to have some statements about how etale spaces and their higher categorical combinations sit inside the simplicial nerve of Top/X. In particular, it should distinguish well etale n-spaces from other representatives of their weak homotopy types, what is not internal notion within the simplicial localization. I do not see how you can avoid at least some set theoretic topology eventually.

]]>not the whole precise adjoint relationship between the internal and external point of view.

Could you remind me precisely what this adjoint relationship refers to?

]]>this theorem about stalkwise weak equivalence

Look at it on the right side of the equivalence:

a weak equivalence of simplicial presheaves on a paracompact topological space is one that is stalkwise a weak equivalence of simplicial sets. If the simplicial presheaves in question happen to be set-valued, i.e. happen so be ordinary simplicial presheaves, then stalkwise weak equivalences are just stalkwise isomorphisms of simplicial sets. So weak equivalences between oo-stacks that happen to be ordinary sheaves are ordinary isos of sheaves.

Now the equivalence tells us that these stalks are the fibers of an etale space over the given point, and hence that the weak equivalences between these etale spaces (over X) are just ordinary isos of etale spaces.

]]>he currently cannot post on the blog

for almost a month already, unless I am not at work (at home it is OK)

I am not getting precisely (just vaguely) this theorem about stalkwise weak equivalence, but anyway the adjoint pair is not only for etale spaces; only the restriction which is an equivalence is. Therefore you might be right that the very etalification can be expressed only as a corollary of the Lurie's theorem, but not the whole precise adjoint relationship between the internal and external point of view. I think that the point-topological information which is lost after taking simplicial localization is lost, even if the etalification for the representing members of weak homotopy classes is faithful as you say. Edit: even more I am not sure if the statement about weak equivalence for the essential image in the Lurie's theorem can easily be shown to have the same weak equiv thus iso property as we know for the usual picture of etale spaces, or more appropriately, the internal n-categories in etale spaces. The identification if it can be done, would resolve big part of my compliant.

]]>That's right. I think Zoran is posting here because he currently cannot post on the blog. But it is true that we should move this elsewhere. Either back to the blog, or to a thread on an nLab entry that discusses this.

]]>I'm not really following this, but I wonder if the "Latest Changes" area of the n-Forum is really the best place to have this discussion?

]]>Are you sure about this? The weak equivalence of an ordinary etale topological space *over* X seem to me to be the ordinary isomorphisms. Since the morphisms have to be over X the weak equivalences in are stalk-wise weak equivalences. Since the stalk of an etale space is a set, this are just the isomorphisms in this case.

Lurie's theorem there achieves: it tells you what an "n-etale space" is

No, it does not. Even for n=1. It just says what is its derived image.

]]>oh, I see, you want the statement for (\infty,n)-sheaves for n > 1. Okay, right, that's something more general.

Am I speaking Korean ?? This is true that I want for higher n, but not only that I want that it is a TRUE generalization, not analogue. So we have to recover n=1 case with full power. Even for n = 1 Lurie's statemet does not cover the classical case. It covers just the weak homotopy type version and forgets about the details of the space. It gets derived from the start.

]]>In a way, Lurie's statement is just a derived shadow of the true statement which should deal with internal omega categories and omega presheaves over Top. There s a general external - internal point of view: internal ctageories vs. presheaves, but here that adjunction gets a more detailed content when we retsrict to stacks and the left hand side can be characterised in descriptive toplogical terms.

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