There are only 48 hours in the day and 9 days in the week however... as someone said!

sure, I know this well. Please don't read my comment as pushing you to do more work. On the contrary: I am just thinking that if you take the time anyway to drop hints here in the forum, then that time is used more efficiently by dropping these hints into the relevant nLab entry and then linking to it.

]]>One of my THINGS TO DO is to set up a Strong Shape Theory entry and to migrate some of the stuff and references from Shape to there. (There are only 48 hours in the day and 9 days in the week however... as someone said!) :-)

]]>more on topic: it would be nice if all such technical details could eventually be mentioned at shape theory.

For instance statements such as

The strong shape of a space was the isomorphism class of the resulting object in Ho(pro-SS)

should go in the Definition-section in the section "Shape theory for topological spaces".

]]>adopting a BrownAHT with minor modifications. (The paper is linked to from the shape theory page.)

Just a technical comment: you can just include the keyword here in double brackets as on the nLab and the software will automatically create a link: BrownAHT

]]>According to the original descriptions of strong shape (or at least those that were nearest the methodology we are using here), which came from Edwards and Hastings at about the same time as I was looking at a very similar theory, the approach was by localising pro-SS with respect to some levelwise weak equivalences (suitably closed up under retraction). This was the ambient category in which things lived. You then took spaces and applied the Vietoris complex construction to get to pro-SS. The strong shape of a space was the isomorphism class of the resulting object in Ho(pro-SS). In the stuff for that time there were various approaches to doing Postnikov etc. I cannot be sure they were all that good and they need at least a glance to check. I had used a slightly different approach to Ho(pro-SS) adopting a BrownAHT with minor modifications. (The paper is linked to from the shape theory page.) There was a second paper giving related homological results then a third doing an obstruction theory. My memory is that that used Postnikov towers. Whitehead towers I seem to remember were much more tricky for my methods so ??????

Beware Edwards and Hastings produce some neat examples where things that are not as simple as you might think!

Another point with their work is that it introduced proper homotopy theory in a new way and THAT is I think an important potential application for the new insights and methods. The application would need quite a lot of working through to interpret the results in a way meaningful to the sort of questions that need answering and might impact on non-compact manifold theory in a neat way. I do not know and will not attempt to work it out here!

]]>Ah, I'm sorry. At some point you (Tim) wrote

how do they compare with the Borsuk shape theory fundamental shape groups etc. of the space.

so I think I meant 'fundamental shape groups'.

Having a read of some shape theory stuff yesterday reminded me how it all came about. In that case, let me go back a step: is there, for the theory of strong shape, an n-truncation procedure for each n, as there is for homotopy types (or even Postnikov towers)? Is there also a co-truncation, i.e. Whitehead towers?

]]>@David: I'm not sure what you want from 'strong homotopy groups'.

I am getting confused again. The homotopy groups (of whatever strength (:-)) are perhaps not really what we need. If you asked for small models of the strong 2-type I would still not know what you wanted but would feel happier!

]]>Obviously the end result should be an n-groupoid, but for now we can focus on the case of a pointed space and look at the autoequivalences of the fibre functor. The trick (and it's a big trick) is to show that these autoequivalences form a pro-n-group. ]]>

how does one define strong homotopy groups? We could do with a page on this :) Also I have some ideas on showing universality, but as someone once said, I had to leave something to do afterwards. Also, there is a general construction for functorial n-connected covers going through sSet and using coskeleton and decalage and back to Top, but it's hopeless to do any geometry with it, as it is \'as big as\' the geometric realisation of the singular complex of the space in question. My personal hope is that we can rather get something like a topological Trimble n-groupoid model for n-connected covers, but it is at the moment beyond my (rather meager) n-categorical skills. ]]>

Don't spend time on what I had in mind, only trivial things mistified by my own confusion&memory.

]]>okay, I need to catch up with your discussion here. Meanwhile I wrote topological localization. I understand that this is not what you, Zoran, had in mind, but maybe it is useful for something anyhow.

]]>I went last year to read 3.3.1 by a rather reliable reference which claimed that Giraud has 2-fully faithfulness in that proposition, and that the equivalence does not hold (I needed equivalence at that time for my purposes). Now you prove me and my informant to be wrong, what I found still strange as I vividly recall seeing the statement to the opposite and thinking of that the whole evening afterwards.

As far as other topologies on the topos you are right I have been misinterpreting their role above. Maybe that is what I was thinking at the time. But this has no repercussions for Urs-s point of view as this is just another 1-topos behind the picture (to emphasise your clear and obvious remark: In general, there is no site whose 1-topos of sheaves is E but whose 2-topos of stacks is the topos of stacks for J on E).

2nd edit: thanks Mike. I think I reconciled most by now, at least with Urs's point of view if not with history of my misunderstanding.

]]>Ohh, I just saw your last comment. Of course, if you look at a topology on the 1-topos other than the canonical one, then you'll get a different notion of stack. But I don't think it makes sense to call those stacks on some *site* defining the 1-topos. Stacks for some random topology J on a topos E are just that, stacks for the topology J on E. In general, there is no site whose 1-topos of sheaves is E but whose 2-topos of stacks is the topos of stacks for J on E.

Edit: Clarified, since I'm not sure if by "regular epimorphism topology" you meant the canonical topology, or the topology generated by singleton (regular) epimorphisms.

]]>In Remarque 3.3.4 on p93 of "Cohomologie non-abélienne" I read

On résume (3.3.3) en disant que, par image directe et inverse, la donnée d'un champ sur équivaut à la donnée d'un champ sur le topos associé .

which I translate as

We summarize (3.3.3) by saying that, via direct and inverse image, to give a stack on is equivalent to giving a stack on the associated topos .

This is what I was saying before. If giving a stack on a site E is equivalent to giving a stack on the associated 1-topos, then two sites having equivalent 1-topoi of sheaves cannot have inequivalent 2-topoi of stacks.

]]>As I wrote in the first message in this chain we can have two different choices of sites which give the same 1-topos of sheaves but different 2-topos of stacks. If I start with a topos, it is USUAL to talk about stacks in regular epimorphism topology. This corresponds to a particular choice of a site producing that topos. This is usual and default choice in topos theory. But it is not necessary. One can take another choice of site, and Giraud claims that it is not necessary that the 2-topos of Grothendieck 1-stacks on that site is equivalent to the 2-topos on the regular epimorphism site.

I will look at your entry, but the question is simple and non-technical.

Edit: I do not know if the nonequivalent sites still appear if we just look at stacks in groupoids, but I would guess it is not essential. I can not find the exact assertion in the Giraud book file at the very moment.

]]>Zoran,

I am sorry for not getting your point. Maybe I still don't get it. You write:

but the stacks on an arbitrary chosen site given the 1-topos.

What do you mean by an "arbitrary site given a 1-topos"? Or do you mean just any arbitrary site? I am confused!

It seems to me that the important point is the one we mentioned before, that for finite n, there is only one localization of n-presheaves on a site that is a "usual localization".

More precisely, by "usual localization" I mean what in HTT is called "topological localization" and the precise form of the statement is proposition 6.4.3.9 there.

I suppose you are asking about "non-usual localizations". In which case I am probably out of my depth. I have as yet very little feeling for n-toposes that do not arise by "topological localization".

Anyway, in order to prepare the ground for further discussion, I will quickly create an entry topological localization.

]]>I am asking about THESE choices, NOT the Grothendieck case. You are giving me repeatedly the obvious answer about the classical Gorthendieck case. I am asking about the puzzling case (I used to ask you about that still in Bonn but I assume you now know the answer), where a nonstandard choice for 2-topos is taken, that is not the stacks in regular epimorphism topology but the stacks on an arbitrary chosen site given the 1-topos. So 1-topos AND 2-topos are given.

Mike: Giraud's book

]]>]]>Sorry, what reference is "Giraud"?

Sorry, what reference is "Giraud"?

]]>what is the infinity topos produced from these choices as opposed to the old Grothendieck case where I have just 1-topos and simplicial object inside ?

If the topos is a sheaf topos, then simplicial objects in it are of course just simplicial sheaves. There are different model category structures on simplicial sheaves. Some represent hypercomplete (oo,1)-toposes, some not. But if the topos has enough points and if one takes the weak equivalences between simplicial sheaves to be stalkwise weak equivalences, then this amounts to producing the corresponding hypercomplete (oo,1)-topos.

]]>Urs I did not understand your answer 50. I you are given a Gorthendieck 1-topos of 1-sheaves and Grothendieck 2-topos of 1-stacks, what is the infinity topos produced from these choices as opposed to the old Grothendieck case where I have just 1-topos and simplicial object inside ?

This should also lead to some intrinsic consistency condition on which 2-topoi are compatible with the 1-topos chosen (classically it means there is a site which can afford both initial data are structures over it, but this is not intrinsic) ?

Mike said

I was not aware of that, nor do I understand how it can be true, since the 1-topos contains all of its sites as generating subcategories, so it seems that a stack on any of its defining sites should be the same as a stack on the big site of the topos itself.

This is in Giraud, in my memory about page 98.

]]>@ Zoran and Mike,

yes, so as we discussed last time elsewhere, if an object satisfies Cech descent, then it already satisfies descent over all *bounded* hypercovers. If the object is itself truncated, then that means it satisfies descent over all hypercovers. So a difference appears only for untruncated objects, i.e. genuine oo-stacks. For these, one finds examples that satisfy Cech descent, but not hypercover desent (as in the appendix of Dugger-Hollander-Isaksen).

But useing general simplicial objects in an ordinary topos means admitting untruncated oo-stacks. So it is a reasonable question to ask which corresponding (oo,1)-topos is implicit in classical topos theory. And since usually the stalkwise weak equivalences are used, the answer is: it is the hypercomplete version. If the underlying topos has enough points. (If it doesn't have enough points, then i am afraid the classical definition has no really good meaning.)

]]>@Tim: you can consider the definition of homotopy groups based on monodromy in every (oo,1)-topos. All I am saying is that it will coincide with "the other definition" only under additional assumptions.

I'd be glad to learn more about what happens in situations that do not satisfy the various conditions used at homotopy groups in an (infinity,1)-topos, such as locally contractible, paracompact, etc.

]]>@Urs I accept that but feel that somehow there is a deficiency in the theory if we cannot approach moderately grotty non locally connected spaces such as the Warsaw circle or limit situations like the solenoids. (Especially these later as they are related to strange attractors etc. and it would be interesting to push this theory of (oo,1)-toposes to look at the geometric style invariants of such spaces. I am probably hoping for too much.)

]]>