Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Lurie, Theorem A.1.1, (I have not found the reference!)
You just need to follow the link given there to find it. It is theorem A.1.1 on p. 134 in Lurie’s article on -algebras.
Meanwhile this is also section A.3 of Higher Algebra.
Lurie proves a homotopy version of the old result (already in Spanier, but much older) that the includion of the U-small singular chains into the singular chains is a chain equivalence. This does not seem to imply the higher dimensional vKTs without more work. Phil Ehlers explored in his thesis the way in which this sort of result but for simplicially enriched groupoids seemed to give a notion of thinness which allowed the retraction of the singular invariants of the whole space to be collapsed down to those of the colimit of the small space. He did not take this idea far enough to be publishable.
Ronnie’s point is I think that whilst his results give for (partial) algebraic models of the homotopy type a calculation of the whole of the model given the parts, the result of Lurie only gives things up to homotopy. This is perhaps the difference between the methods of Combinatorial Homotopy theory and Homotopical Algebra. Both are important, but it would be interesting to see if for instance the Brown-Loday vKt could be derived from Lurie’s result without a large amount of extra effort. (I have considered the problem yet, so have no great light to shed on it!)
A more specific question would be to see what results as Lurie proves say about the homotopy type of the suspension of a K(G,1) as there the Brown-Loday form allows explicit calculations.
There is no evidence from what is written that he can recover from his general theorem his own statement of the SvKT
It’s a direct consequence of lemma A.1.3.
Ronnie’s point is I think that whilst his results give for (partial) algebraic models of the homotopy type a calculation of the whole of the model given the parts, the result of Lurie only gives things up to homotopy.
If you look at the proof (p. 136) you see that it proceeds by computing the 1-categorical colimit that you are looking for.
Concerning the entry higher homotopy van Kampen theorem: what I did suggest is that if you do edit the entry you are welcome to add useful information, but I ask you please not to continue a fight against other authors in homotopy theory. If somebody else gives a construction and proves a result that generalizes yours, this is not belittling your contribution, but on the contrary, shows that is is useful, fruitful and blossoming.
On the face of it, without intensive thought, the way to connect the result dealing with homotopy colimits of singular sets and strict colimits of fundamental crossed complexes is a vKT for the functor . This probably is a serious amount of work, as it looks like a serious theorem to me. Notice that because strict colimits appear it is not automatic by usual homotopical abstract nonsense.
An interesting question is this: what is the homotopy colimit of the simplicial sets associated to strict algebraic models of homotopy types. That is, we are just asking whether the (EDIT: essential or possibly up-to-homotopy) image of is closed under homotopy colimits. A separate, but related question, is when we can write these homotopy colimits as strict colimits. Given a good relation between the two theorems, as in my first paragraph, we can then say that Ronnie’s results are of this form.
I should point out that the version that appears in the nLab page is not the same as appears in appendix A in Higher Algebra. The latter is closer in flavour to Ronnie’s work involving filtered spaces, rather than the version involving pushouts of open sets. The nLab page of HHvKT is the target of a link at the page for Higher Algebra under the heading of appendix A. At the very least the relation to the exit path version should be clarified (yes, yes, I know, I should do this rather than say ’someone should do it’…). I don’t know whether this is also a non-trivial result, but it seems pretty serious.
@Urs Ronnie is NOT ‘fighting other authors in homotopy theory’. I think you may not understand the viewpoint that he is defending.
The fact that the singular complex of a space is the colimit of the U-small singular complexes under certain conditions has been well known for a long time. Lurie’s proof is neat and concise… fine. Within homotopy theory the calculations tend to want algebraic models of things. Perhaps for instance with the 2-type and their 2-groupoid models, this will be up to an algebraic form of homotopy (and that homotopy can be made explicit). One wants smallish calculable invariants and perhaps eventually numbers, explicit cohomology classes etc. The classical SvKT can be derived immediately from Lurie’s statement since the fundamental groupoid is a left adjoint and is homotopy invariant, but the Brown-Higgins’ 2-dimensional one is surely not immediate from Lurie’s form. The point being that their specific model for a 2-type depends on a presentation of the homotopy type of the space, e.g. as a CW-complex. With that information you get a definite representative for the homotopy 2-type of the space and a recipe for calculating that specific representative from the analogous small models. Using the singular complex you get enormous models for the homotopy 2-type. Those can be useful, but are not the same. In some of the current writings on (weak) infinity groupoids.
This is a complementary vision of some parts of homotopy theory (and is well represented in modern homotopy theory which is not just model category structures). As I have said before this is nearer the idea of Combinatorial Homotopy Theory as put forward by JHC Whitehead than the homotopical algebra stream of research.
I’m with Ronnie and Tim on this one. Also, the very idea of calling the total singular complex of a space its “fundamental ∞-groupoid” borders on ridiculous. The fundamental ∞-groupoid of a space should be a Batanin-Leinster or Grothendieck-Maltsiniotis weak ω-groupoid. Declaring the singular complex to be the fundamental ∞-groupoid just feels cheap, since we end up with something that is, for all intents and purposes, as difficult to work with as the space itself.
I should point out that the version that appears in the nLab page is not the same as appears in appendix A in Higher Algebra.
It is, in section A.1 of Higher Algebra. The statement in A.3 is the generalization to stratified spaces and “exit paths”. And I don’t think that this stratification can be identified with the filtering used elsewhere.
The filtering is essentially a way to remember cellular structure even when talking about topological spaces. The stratification gives genuinely more structure: it leads to fundamental higher categories.
The classical SvKT can be derived immediately from Lurie’s statement since the fundamental groupoid is a left adjoint
And so are all the higher truncations, say to the fundamental 2-groupoid.
I would suggest that if you want to have the theorems understood as being about models, and not about the intrinsic statement that they present, then let’s start a new entry models for higher van Kampen.
ridiculous
By the way, I will not engage anymore in discussion involving this behaviour. You do notice that I could just as easily state the claim that your point of view is ridiculous. And then, where would we be? This is a forum for discussion, and discussion ends when the participants don’t stick to basic rules of intellecual discourse.
there is a world of difference between a homotopy colimit of lax structures and a strict colimit
Lurie’s result is about strict (1-categorical) colimits. That’s the point.
I think the basic disagreement is what constitutes a higher vKT. Lurie’s result could equally well be called the homotopy small simplex theorem (a good name) or the simplicial approximation theorem, (:-)), which would be a good name if it had not already been used somewhere else! What’s in a name… except the point of the difference in interpretation can be important. (I should point out that there are other vKTs around, for instance, Artin and Mazur had a paper in Topology in about 1970 with van Kampen in the title. How does that fit in?)
What I am interested in is not so much the name but what can be derived from these results. I know that the truncations are left adjoints (at the suitable n-level) but then calculations in Ronnie’s various vKTs gives strict colimits not homotopy colimits and the universal property in Lurie’s statement is a homotopy colimit even if the construction is a colimit (???) May be I am being thick, but I still feel that we have two different threads of development here.
An example of where Lurie’s result does not seem to give very much information is in applications of these sort of ideas to the work on higher generation of groups (Abels and Holz) and more recently on complexes of groups. There the local information is partial giving a bit of a resolution of a group and the nerve of the indexing category is what glues things together. The various parts are constructed via combinatorial methods and the homological group theorist hopes to gain information on the resulting colimit or homotopy colimit, comparing it with resolutions of the big group. You do not use the singular complex here, rather it is CW-complex techniques that count, and that is more combinatorial and infinity groupoidal in nature.
There may be a gap in the developed theory, namely the study of the combinatorial / algebraic / categorical models within a homotopy type. At a naive level, looking at the models e.g. crossed complex ones, of some homotopy type represented by a CW-complex and now performing subdivisions on the complex structure. I do not know of much work in this area. It would be relevant for my work in HQFTs so if someone knows something on this please tell me.
@Urs, appendices: A1 vs A3
ah, I see. I was thinking that the stratified space stuff linked in with Ronnie’s work on filtered spaces and strict models associated to them. Perhaps a groupoid version does, and would be interesting for further discussion at some other time.
It is interesting to follow through in Spanier to see where he uses his small simplices argument. it is (p.188) in a proof of excision and is related to Mayer-Vietoris.
I feel there is a different aspect coming out in the old nice objects vs nice category dichotomy, namely strictness. The homotopy category is defined a priori in terms of topological spaces, or even CW complexes, and Quillen tells us that simplicial sets do the job too. The latter are a topos so form a nice category, whereas the CW complexes are nice objects, but form a category that is not up to scratch for high-powered category theory. However, we can trade off some of the ’badness’ of the category of CW complexes and keep the geometric combinatorial structure in passing to for example the category of crossed complexes, but lose homotopical information. The category of crossed complexes is very nice (aren’t they models of a finite limit sketch?) and the objects are very nice indeed, being algebro-combinatoric in nature. More importantly, we find that we retain some information that is lost in merely passing to e.g. homotopy groups.
There is an n-category cafe posting discussing the relative merits of algebraic vs ’geometric’ (or non-algebraic) definitions of homotopy types and the homotopy hypothesis, and simplicial sets (or Kan complexes) fall exactly in the middle of the spectrum presented here. I can’t find it at the moment (this and this are similar), but there was discussion involving Nikolaus’ algebraic Kan complexes as falling on the slightly algebraic side, and the various ’exist a filler with contractible space of choices’ versions on the non-algebraic side.
There was a tongue-in-cheek remark that if one defines a space as a Kan complex and an -groupoid as a Kan complex, then the homotopy hypothesis is trivial, if that helps anyone else find it…
Ah, this is it, with comments by Urs and others on considering simplicial sets/Kan complexes as higher categories/groupoids starting at this comment. I guess this may be why Urs doesn’t feel like debating this again, as he has done so at length in the past.
Danny Stevenson kindly provided further references, which I have added here.
@Ronnie: Can you add in the reference to the vKT for cat^n groups? I think it is important because it leads to new tensor product type constructions as well as the possibility of calculation. I could add this but you probably have some pre-existing text that can be adapted.
The debate above suggests that the entry on higher vKTs really needs splitting into the topological singular simplex version and the Generalised vKTs of Ronnie and Phil Higgins, and Ronnie and Loday. Perhaps the topological version then needs a discussion on its origins (small simplices, classical chain equivalence results as in Spanier etc.Does any one know the origin of that small simplices results, e.g. is it Mayer-Vietoris, excision or where? I ask because to show the continuity of the modern result with its classical antecedents puts the modern theory in context.)
@Ronnie: The Batanin-Leinster or Grothendieck-Maltsiniotis fundamental -groupoid is indeed a generalization of the fundamental -groupoid. The strictness of the fundamental -groupoid comes not from the “fundamental groupoid” part of the functor, but instead from the collapse functor.
By the way, if I am seeing correctly then the section “A.3 Seifert-van Kampen Theorem” in Higher Algebra was replaced by a new version just today, pp. 845.
@Ronnie,
such a construction (fundamental groupoid on basepoints), can be seen as as pullback of and where denotes the codiscrete or banal groupoid. This is what is also known as an induced groupoid along the inclusion, or, as I prefer to call it, the base change along the inclusion.
There are higher groupoid versions of this, which arise from the n-ary factorisation system on (n+1)-groupoids (essentially the Postnikov-Moore tower), together with (Grothendieck) fibration-like lifting properties, and this would help, I imagine, to express relative homotopy (n-)groupoids. I should write down a version of the above for the triple and the fundamental bi-/2-groupoid - I’ve thought about this before, but not finished it off. (In fact I was trying to construct a span of weak equivalences of topological bigroupoids between the topological Hardie-Kamps-Kieboom-Stevenson fundamental bigroupoid and a suitably topologised version of the fundamental strict 2-groupoid, both of a CW complex with its usual filtration. Never quite finished it off…)
Later:
ok, here are some sketch details. Let be a filtered space, and be the relative path space where the set is mapped into the subspace . Then let be the groupoid with object set , arrow set and a unique 2-arrow between any two parallel 1-arrows, and be the analogous construction but for the relative fundamental groupoid. Let be the fundamental bigroupoid of the space (forgetting filtered structure) and be a fundamental 2-groupoid, when this can be done (e.g. Hardie-Kamps-Kieboom give one in the first paper on this). There is a 2-functor (analogously for the strict case, which I will leave implicit), and also a 2-functor . The (strict!) pullback of these two 2-functors gives a fundamental bi-/2-groupoid for the filtered space
[I’ve squashed some steps together in the above paragraph, so it doesn’t generalise cleanly in the form given, but it should be possibly to do an inductive construction, working up from the objects, using the n-ary factorisation system as I said above.]
Given a CW complex with its usual filtration in the bottom dimensions we can take the weak version and I think strictify it by identifying paths which are homotopic inside . I’m not 100% sure how to describe this in an abstract way (yet), but it would probably involve the coequifier of a pair of natural transformations between . (This sort of construction one would do inductively from the top-dimensional arrows down if trying to generalise to higher dimensions.)
Apart from that, all of the above internalises nicely to () once we have topological (smooth) fundamental bi-/2-groupoids, since we are only talking about finite limits (and in the manifold case we find we have submersions where necessary).
Ideally, one should be able to write this down in a model-independent way, so that given a definition of weak -/-groupoid we can form these pullbacks and weighted -colimits using abstract nonsense to get the result.
Calculations on the other hand…
Hi Ronnie, more discussion at the Cafe you might like to read. Anyway, you can see we’ve been over this ground a lot. The best is to extract actual mathematics from the discussion, rather than get mired in details like ’what is a good model of an -groupoid?’ based on vague (or even not so vague) desiderata for such a theory.
I only just catch this here from #25, as I have not been following all exchanges:
Thus it seems Urs was convinced that the result quoted was a generalisation, simply by the utilisation of the name.
No. Instead I read the result and its proof, understood it, compared it to other things that I understand, and drew a conclusion. That’s how I usually do math, I hope: not by agenda but by understanding.
Generally: I find this entire discussion here would benefit from moving away from discussion of tastes, feelings and insinuations and instead sticking strictly to the math.
I think I’ll try computing using the filtration (east and west ’poles’ and equator), my construction above, and the coequaliser , just as an exercise…
A lot of this discussion seems too confrontational to me. Sometimes one wants tools that are well-adapted for concrete calculations. Other times one wants tools to prove general abstract theorems. Sometimes abstract theorems yield concrete calculations or suggest calculations that one should do; other times concrete calculations point the way towards new abstract theorems. Some of us are more interested in, or better at, one side of the coin; others prefer the other side. I’m sure we all know this.
It seems to me that we are probably looking at multiple different ways to generalize the classical VKT which are not reducible to each other (at least, not obviously). One generalization that is more 1-categorical may be more useful for computations; another, more, -categorical one may be more useful for abstract purposes.
Can I put in a request, though, that the word “lax” be reserved for structure that is preserved up to a not-necessarily-invertible transformation, as is now standard in category theory, with “weak” or “pseudo” used when the transformation is invertible? (Which seems to be the situation being considered here, is that right?)
About Lurie’s Higher algebra. The date on the frontpage of it gives May 18, 2011? Is this a new version?
I think that the relationships between the Lurie result and the generalised vKTs of Ronnie et al could do with being made precise. As Ronnie says Lurie’s statement is closely related to the result in RB-PJH-RS’s book on page 363. There are a whole lot of related results (excision, Mayer-Vietoris, vKTs, small simplices) and perhaps some clarification is needed (not just in the nLab!) on the precise relationship between all the various named forms. I do think that the present situation could be confusing for a beginner.
Re #39, Mike’s subtle account of possible interrelations between more abstract and more computational approaches puts me in mind of Hilbert and Gordan on invariant theory. If you haven’t done so already, I thoroughly recommend that you have your preconceptions about that episode blown out of the water by reading Colin McLarty’s Theology and its discontents: David Hilbert’s foundation myth for modern mathematics.
Here’s an attempt to deduce the usual SvKT from Lurie’s A.3.1:
Take to consist of three objects, corresponding to the open sets , , and their intersection . The condition (*) of the theorem is equivalent to . The conclusion of the theorem says that the -groupoid is the homotopy pushout of and along . Now, apply the 1-truncation functor which is left adjoint to the inclusion of 1-groupoids into -groupoids. Note that the 1-truncation of is the fundamental groupoid of . Thus we obtain that is the 2-pushout of and along . In fact this 2-colimit is a strict colimit, because the maps we’re taking the pushout of are injective on objects. So this recovers the classical SvKT as well as computes .
If you have a cover of by arbitrarily many open sets, you can take to be the diagram of all finite intersections. In this case (*) holds because the fibers are nonempty co-filtered posets, and you get the same result (again the 2-colimit is strict, because for any open , the map from the strict colimit of all , , to is injective on objects).
Given a set of base points , you can base change the whole situation to . Base change preserves homotopy colimits so you get as a (strict) 2-colimit of groupoids. It doesn’t seem like we need to intersect anything.
Ronnie, I think your question is entirely justified (and I think Marc has just given an excellent answer). But I did feel that some of your comments sounded disparaging towards abstract theorems that are not directly calculationally useful (just as some other remarks seemed unjustifiably dismissive of your question). If that was not your intent, I apologize.
@Ronnie: The last part of my post about “base change” is indeed nonsense! But at least I hope I’ve shown that the classical SvKT at the beginning of Lurie’s A.3 follows from his “more general” theorem, as does the computation of .
What I can say for sure is that the additional conditions when there is a set of base points have nothing to do with the relation between strict and 2-colimits: the strict colimit will always be a 2-colimit for the reason I gave in #44. I would guess that the condition on 3-fold intersections has to do with the 1 in ; for there would be a condition on 4-fold intersections, etc.
Of course, it is important to find out what a given theorem can and cannot do! But I do think that it is unnecessarily confrontational to assert that when someone proves a theorem with one purpose in mind, it is “his job” to figure out how that theorem relates to other theorems that other people have proven with other purposes in mind. Certainly, someone should do it, but everyone is busy. It seems to me that often this sort of thing only gets done when someone else comes along and needs it for something (or is just curious enough, or bored enough, to put in the work).
I am wondering if there is not a need for a ‘local-to-global’ entry that chats about the general SvKT and the small simplex theorem (both the classical one for (co)homology / chain complexes and various of the others (i.e. homotopy results) including Lurie’s) and then links to descent.
Another terminological suggestion is that the Lurie’s result be called the homotopy SvKT /homotopy small simplex theorem, whilst the various versions of the vKT due to Ronnie are called the strict generalised vKTs. I know Ronnie likes to call the Brown-Higgins results the higher homotopy vKT but there is also the Brown-Loday generalised vKTs for the n-types. The specific versions can then be referred to as the crossed complex vKT etc.
One key point, if I remember rightly, of the proof of Ronnie’s crossed complex vKT is the fibration theorem which relates the filtered space singular complex to the corresponding omega-groupoid. The crossed complex model is more-or-less a quotient of the singular complex model with nice conditions on the filtration giving nice properties on the quotient map.This suggests that the use of thin element structures in this context is the key to understanding the relationship between the two types of result. I have not thought out how this might manifest itself in the Brown-Loday case, but my intuition is that this may be worth doing, i.e. that analysing the relationship between the weak infinity-groupoid models of a homotopy type and the various strict models of the n-type, the crossed complex models, and so on, then applying the resulting ‘machine’ to homotopy colimit result of Lurie, after first restricting via pullback etc to the filtered case.
That idea relates in turn to various points about the intermediate models, e.g. we have not got a vKT for 2-crossed complexes and that might be useful for various calculations. I must stop now, but if no one objects or has a better suggestion i will try to start reorganising things along the lines I suggest in the next few days.
FWIW, I agree that the current statement on the nLab page is misleading; it doesn’t seem to be clear whether one can derive your strict van Kampen theorem from the weak -groupoidal van Kampen theorem cited above it. It would be nice if someone could write out a comparison in dimension 1 along the lines of Marc’s suggestion (which seems at least likely to not be too difficult); making any comparison precise in dimension sounds like a much more significant undertaking.
I have pencilled that in ‘to be looked at’ later this summer (if summer it is… rain, wind, clouds… , but a good ’feu d’artifice’ for the 14 juillet in Paris, with no rain!).
I posted an answer to the MO question.
It seems to me that from the nPOV, the SvKT “factors” into three parts:
Lurie’s general theorem is (1), while (2) is basically abstract nonsense. But for concrete computations, of course (3) is crucial! And (3) is also the most difficult to generalize to . I suspect that from the nPOV, one could say that the value of cubical things / crossed complexes / etc. is that they are a sort of strict presentation where (3) holds usefully. (Which is not to denigrate other points of view on other values of such things… or to say that there may not exist other useful strict presentations… and could be entirely wrong because I still haven’t really understood the cubical approaches myself (sorry Ronnie).)
So which part deserves to be called “the higher homotopy SvKT”? I dunno… for me, it’s certainly (1)+(2) that carry the “intuitive” meaning of the SvKT as I learned it. I kind of like Tim’s suggestion of “homotopy SvKT” and “strict SvKT”.
Lurie’s general theorem is (1), while (2) is basically abstract nonsense.
In Lurie’s theorem the colimit is strict. If it is not clear from the statement of the theorem, search the proof for where it says “projectively cofibrant”.
Then he should state so in the theorem, namely “Sing(X) is the homotopy colimit, and even the strict colimit, of the diagram…”. :-)
In Lurie’s theorem the colimit is strict.
The colimit in his proof of Prop. A.3.2 is a strict homotopy colimit. But the colimit in the proof of Theorem A.3.1 is not (at least, not as far as I can see).
However, you make me realize that my part (3) should also emphasize the use of an algebraic strict model, in order to really capture the computational value of an SvKT.
The colimit in his proof of Prop. A.3.2 is a strict homotopy colimit.
And that’s the one we need for the traditional theorem: the one that takes the colimit over a cover.
you make me realize that my part (3) should also emphasize the use of an algebraic strict model, in order to really capture the computational value of an SvKT.
One can postcompose with the Quillen left adjoint that forms algebraic Kan complexes.
You will maybe say that this is no the kind of “algebraic” that you actually mean. I’d think for the wish-list to be useful in item (3), one should provide some precise idea of what “computational value” is and why your preferred model has more of it than somebody else’s preferred model. But usually the most computational value is in having as many diferent models as possible.
And that’s the one we need for the traditional theorem: the one that takes the colimit over a cover.
It depends on what “traditional theorem” you are looking at. In my MO answer, I didn’t see a way to get Ronnie’s version with a specified set of basepoints out without using an auxiliary category like in A.3.1. But maybe you do?
one should provide some precise idea of what “computational value” is and why your preferred model has more of it than somebody else’s preferred model.
Thankfully, in this case we can be very precise about this. The traditional SvKT allows us to compute the (0-)group of a space using 1-categorical colimits in the 1-category . I don’t see that in algebraic Kan complexes.
Ronnie’s version with a specified set of basepoints
I haven’t thought about that version. I’d tend to regard a choice of basepoints as something one should avoid, unless we are in a context where it is part of the data, such as stable homotopy theory.
The traditional SvKT allows us to compute the (0-)group π 1 of a space using 1-categorical colimits in the 1-category Grp.
Or equivalently in .
I don’t see that in algebraic Kan complexes.
One generalizes a 1-pushout of groupoids to a 1-pushout of -groupoids, and these can be chose to be algebraic, if desired.
But let’s wrap this up: I am glad that we finally got to the point where we have all opened Lurie’s proof and seen what it actually says, notably that (and where) it involves 1-categorical limits.
Now we can get back to the very beginning of the discussion and see how to improve the entry higher homotopy van Kampen theorem. Do you (some of you) still think that it “distorts” the situation?
I don’t think it distorts the situation, but – as with essentially all Lab entries – I see big potential for it to be expanded. Maybe somebody feels like doing so.
The entry does not yet ’do justice’ to the variety of generalisations of the classical SvKT available, and perhaps should possibly indicate where there is work to do to clarify the relationships between them. I do think that we should use ’small simplex theorem’ as an equivalent term for Lurie’s version a bit more.
I mean to have a go at reworking the entry and adding new entries on the Brown-Loday result, but this needs time that I cannot put in just at the moment. Perhaps ’ideally’ this should all be linked into the local-global / descent framework. I really should include some of this in the Menagerie as well! (so I do feel like doing it and hopefully in the not too distance future… .)
I started reworking the page, but realized that most of what I had to say was more about the ordinary van Kampen theorem, so I did a bunch of editing there. What do people think about that page now? If we agree that it’s better (or at least not worse), then I or someone else can edit higher homotopy van Kampen theorem along the same lines (although using Tim’s suggested terminology, perhaps that page should just be called “higher van Kampen theorem” since it is about both homotopy and strict versions).
Mike: put it in the plural: "higher van Kampen theorems" ?
I suppose. Generally we don’t use plural page names, of course, but I guess the situation here is kind of different.
I have had a go at improving the higher SvKT entry. Please have a look and give me feedback.
O my, I thought we were talking about the page higher homotopy van Kampen theorem. I didn’t realize there was also higher van Kampen theorem! We should probably merge them…
The higher homotopy …. page looks like an earlier version of the higher vKT page so perhaps merger is not necessary just redirecting, renaming or something. I cannot check on how much they are the same as I am just on a laptop and have no printer available (I know there are ’modern’ ways of comparing but the old fashioned way works well!)
On content, I seem to remember some other types of vKT in the literature but have not looked for them yet.
I know there are ’modern’ ways of comparing but the old fashioned way works well!
But didn’t you just say that the old fashioned way isn’t working for you at the moment? (-:
The page higher homotopy van Kampen theorem does indeed look almost like an exact duplicate of version 9 of the other page. I wonder how that happened? We should delete “higher homotopy van Kampen theorem” and redirect it. But I have no time to do so at the moment…
It had already been done (about three times!). The redirects at the bottom of the source were repetitious, so I did some tidying. The other higher homotopy vKT page is still showing up… cache bug?
I forgot to ask. Does anyone else know of places where there is some form of higher vKT?
1 to 73 of 73