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created simplicial homology (instead of cellular homology)
Is it worth putting in a historical note that simplicial homology applied to simplicial complexes originally, etc. and via that to spaces using triangulations?
Cellular homology is a term used for the homology of CW-complexes based on their cell decomposition, which is important when looking at the chains on the universal cover, for instance, and thus in simple homotopy theory. I have had some thoughts on the nPOV version of simple homotopy and what it might mean, but they have never been pushed far enough to warrant putting them down here, but I would be interested if anyone else has thought about this.
Tim’s right: there is this cellular homology theorem which is a sometimes-useful tool in computing homology of a CW-complex. I mean the one described here. (Which reminds me that we have virtually nothing at the nLab on spectral sequences, just a mention on one of Urs’s pages.)
I have created a stub on cellular homology. I added a terminological note to simplicial homology and a reference to Hatcher’s book.
we have virtually nothing at the nLab on spectral sequences
I thought spectral sequence is at least giving a decent overview by now. And at spectral sequence of a filtered complex quite a few details are spelled out by now.
I’ll come back to these entries later this month.
I am at a loss! I must be going crazy; I could swear I entered “spectral sequence” into the search function, and came up with only what I said before. My apologies!!!
Query: is a CW-complex a space together with a CW-decomposition, or is it just the space together with the knowledge that a CW-decomposition can be given? Interesting question linked to Jim’s comment. We seem to have opted for the second of these. We then have no entry on cellular map so the category of CW-complexes is not really defined in the nLab. (Do we want Cw-complexes and cellular maps, or CW-complexes and continuous maps, as the category… or both? We do have the cellular approximation theorem. I will have to see what the various sources say, but only have some basic and old texts here with me, so any thoughts anyone on how to get a neat set of entries? My preference would be to add cellular map to CW-complex rather than to start a new entry.
I assumed that the default meaning of “category of CW complexes” is: the category of spaces which admit a CW presentation, and continuous maps between them. (Edit: I have edited the page cellular approximation theorem so as to refer to chosen CW presentations.)
A constructivist would probably say a CW-complex should be equipped with a CW presentation, even if in defining the category of such one chooses to ignore that structure.
Also relevant to “the category of cell complexes” is this paper.
@Karol. That agrees with my viewpoint, I think. Can you adapt / edit what you have written and add it at a suitable place in the entry on CW complexes? If we use CW-complex as a shorthand for ’Space admitting a CW-complex structure’, as seems to be the case, then we probably need a note to say what you are saying. (The same point could be made about manifolds, orbifolds, etc. We don’t always carry around the atlas, i.e. the presentation of the object when working with it.)
Karol, that sounds good to me too.
Thanks for the edit. I might be the one responsible for “CW complex”; I don’t remember. I don’t think it’s too serious because “CW-complex” redirects there; if you’d like to change the title, go ahead, but beware of the dreaded cache bug!
beware of the dreaded cache bug!
So please don’t do it right now. Because “bewaring the dreaded cache bug” means that I need to go, log into the server and do this and that. No time right now.
@11, that seems to me like a bizarre midpoint between property and structure to choose. Why not define a CW complex to be equipped with a full presentation in terms of cells? (Also, I don’t see why to prefer “CW-complex” over “CW complex” – we say “cell complex” not “cell-complex” usually…)
It may seem bizarre at first, but this is exactly how it is used in practice. Most of the constructions you can do with a CW-complex depend on the choice of skeleta but not on the choice of attaching maps. For example in order to define the cellular complex we surely need a filtration. However, the choice of specific attaching maps corresponds to the choice of bases in the groups of cellular chains. We usually don't want our free modules to come equipped with bases, so we shouldn't insist on our CW-complexes to come equipped with attaching maps. On the other hand if we want the cellular complex functor, then we need skeleta as a structure.
There is also homology of $\Delta$-complexes (as Hatcher calls them) which is in between homology of simplicial set and of simplicial complexes. In fact it is just the homology of semi-simplicial sets (i.e. without degeneracies)
and Brian Sanderson and Colin Rourke wrote a large number of papers on applications of $\Delta$-sets and their applications in geometric topology.
Just because you don’t usually use some part of a structure doesn’t mean it makes sense to leave it out of the definition.
I was referring to the terminology only. There did seem geometric reasons in Rourke and Sanderson’s work as to why degeneracies caused problems. There is probably a newer POV that would get around their objections however just as orbifolds allow one to work with some of the singularities that arise in studying manifolds. It was not simply a generalisation of simplicial sets, but was needed as such in their work.
I guess Mike’s remark referred to my post (18). If that’s the case, would you apply the same argument to the notion of free modules? Would you prefer “a free module is a module equipped with a basis” to “a free module is a module that admits a basis” based on “because you don’t usually use some part of a structure doesn’t mean it makes sense to leave it out of the definition”? And even if you insist on keeping the attaching maps as the structure you run into a similar problem. Namely, how would you justify that morphisms preserve some part of the structure but not all of it? (That’s provided that you agree that cellular maps are right morphisms.)
In the end, I agree that such a definition looks somewhat awkward, but I think that this is the one that is closed to the actual usage.
@24 Karol: There may be confusion here as Mike’s comment could also apply to using $\Delta$-sets instead of simplicial sets.
By the way, some authors (for example Fritsch and Piccinini in Cellular Structures in Topology) use the name presimplicial set instead of semi-simplicial set or $\Delta$-set. I think it’s a nice name which avoids all potential ambiguities.
I had not remembered that term, but you are right it is nicer that $\Delta$-set and less ambiguous than semi-simplicial set, since as Jim pointed out that was initially the name for what we now call simplicial sets.
Would you prefer “a free module is a module equipped with a basis” to “a free module is a module that admits a basis”…?
Yes.
Namely, how would you justify that morphisms preserve some part of the structure but not all of it?
When you define a category, you get to choose the morphisms in it, with no justification necessary other than that they are the useful ones. Otherwise, how would we get the notion of continuous map between topological spaces? Of course, some kinds of structures (notably, algebraic ones) come with an “obvious” sort of morphism which is often the useful one, but there’s no law that we can’t use a different kind of morphism if we prefer.
less ambiguous than semi-simplicial set, since as Jim pointed out that was initially the name for what we now call simplicial sets.
By now, I think the only people who know that simplicial sets used to be called “semi-simplicial sets” are the small number of people left who remember that usage personally, plus those of us who are forced to listen to them constantly bringing it up. (-: It’s certainly a fact of historical interest, but nobody actually does call them that any more, so why can’t we all just forget about it and use terminology in a sensible way now?
If you ignore ancient history, then saying “semi-simplicial set” to mean “simplicial set without degeneracies” is actually less ambiguous than “presimplicial set” or “$\Delta$-set”, because “semi-” is used in other contexts as well to mean “without identities” — semicategory, semigroup — and so a reader might be able to guess what it means without being told. With “presimplicial set” or “$\Delta$-set” there is no hope of that.
And finally, if you actually go back and read the original original paper, you find that “semi-simplicial set” was actually first used to mean exactly this: what we now call a “simplicial set without degeneracies.” (The “semi-” referred there to how these objects different from simplicial complexes.) What we now call a “simplicial set”, with degeneracies included, was then called a “complete semi-simplicial set”. That was a mouthful, and it became clear that the ones with degeneracies were more useful, so people progressively dropped first the “complete” and then the “semi”.
That historical point is very well made. The important point, in any case, is to use a term consistently and, in some prominent place, to explain why it is so used, possibly with a comment along Mike’s line about ’complete semi’ … included for fun. I would therefore support semi-simplicial as equalling ’without degeneracies’.
Would you prefer “a free module is a module equipped with a basis” to “a free module is a module that admits a basis”…?
Yes.
I’m really surprised. Would you mind explaining why? I was under the impression that most mathematicians (and especially those categorically minded) agree that we shouldn’t choose bases and carry around those choices as a structure. Consider this example. Let $A$ be a subgroup of $\mathbb{Z}^2$ consisting of those elements $x$ that satisfy the equation $3 x_1 - 2 x_2 = 0$. According to my definition $A$ is a free $\mathbb{Z}$-module. According to yours it isn’t until I fix a choice of a basis and such a choice would have to be arbitrary. I would like to be able to call $A$ a free module without being forced to make this choice, simply because this choice superfluous for most purposes.
When you define a category, you get to choose the morphisms in it, with no justification necessary other than that they are the useful ones. Otherwise, how would we get the notion of continuous map between topological spaces? Of course, some kinds of structures (notably, algebraic ones) come with an “obvious” sort of morphism which is often the useful one, but there’s no law that we can’t use a different kind of morphism if we prefer.
Part of the problem is that “preservation of structure” is an informal notion. I’ve learned to think that “being continuous” means exactly “preserving the structure of a topological space”, but you are right that this rather arbitrary. On the other hand I’ve always found it useful to think that “structure” is “whatever is preserved by morphisms”. Even if this not a formal notion I use it as a guide helping me decide what should be a structure and what should be a property. For example the notion of continuous map is not the “right” notion of morphism of metric spaces because it preserves far less structure than a metric. I guess this is part of the reason why we do topology in the category of topological spaces and not in the category of metric spaces and continuous maps. I would even go further than that and say that when we do topology using metric spaces, then we are working in the category of metrizable topological spaces and not in the category of metric spaces and continuous maps. Those categories are of course equivalent, so the difference is perhaps purely psychological, but the former choice seems to better reflect which features of our spaces are used as a structure an which are used as a property in practice.
(Full disclosure: I’m trying as hard as I can to argue for a certain perspective here; it may or may not be what I actually think. I’ll decide what I think when I see how the discussion concludes.)
According to yours it isn’t until I fix a choice of a basis and such a choice would have to be arbitrary. I would like to be able to call A a free module without being forced to make this choice, simply because this choice superfluous for most purposes.
But if you define the category of free modules in such a way that its morphisms do not have to preserve bases, then the choice of basis is determined uniquely up to unique isomorphism. So it is not really a “choice” at all and does not even need to be mentioned, any more than when we have a category with cartesian products we always have to write “choose a product of $A$ and $B$ and call it $A\times B$” whenever we want a product.
For example the notion of continuous map is not the “right” notion of morphism of metric spaces because it preserves far less structure than a metric.
I object to the idea that there is a single “right” notion of morphism between any sort of structured object.
It may be interesting to note, once again, that Ehresmann named categories by reference to their morphisms not their objects. This is often a very useful way of avoiding excessive notation as well as agreeing with the view put forward in Mike’s comment.
Just as a side-note, there are indeed a number of notions of morphism between metric spaces: contractive or short maps, Lipschitz maps, uniformly continuous maps to name three more, and I don’t think any of them can make exclusive claim to being “the” right notion.
(Full disclosure: I’m trying as hard as I can to argue for a certain perspective here; it may or may not be what I actually think. I’ll decide what I think when I see how the discussion concludes.)
I could say the same thing. In the very beginning I was quite convinced that I have a very precise opinion on this matter, but the feeling diminishes as our discussion progresses.
But if you define the category of free modules in such a way that its morphisms do not have to preserve bases, then the choice of basis is determined uniquely up to unique isomorphism. So it is not really a “choice” at all and does not even need to be mentioned, any more than when we have a category with cartesian products we always have to write “choose a product of A and B and call it A×B” whenever we want a product.
I also don’t like having my categories with products equipped with product functors. It seems to me that we believe opposite things for exactly the same reason. I believe that if some piece of data is determined uniquely up to unique isomorphism, then it is unnecessary (and somewhat inelegant) to fix a particular choice of this data. You seem to believe that there is no harm in fixing one for the same reason.
I object to the idea that there is a single “right” notion of morphism between any sort of structured object.
I wasn’t trying to argue this at all. I recognize that we are allowed to define both objects and morphisms in whatever way we like and (as soon as the axioms are satisfied) we get a perfectly good category regardless of whether there is a close connection between the definitions of objects and morphisms. What I was trying to argue is that there is some set of informal guidelines that tell us how to define categories in a “neater”, “more elegant” or “more efficient” way.
It may be interesting to note, once again, that Ehresmann named categories by reference to their morphisms not their objects. This is often a very useful way of avoiding excessive notation as well as agreeing with the view put forward in Mike’s comment.
But this is exactly where I was coming from. Those are the morphisms that dictate the nature of objects, not the other way round. And this is the main premise of those “informal guidelines” I mention above: the structure of objects should be defined to fit as closely as possible to the notion of morphism that we are interested in. I do not insist that everybody follows those guidelines, but just from looking around and seeing how people use category theory I gathered that this is what most mathematicians do anyway. So I am surprised to see a strong opposition to this idea.
Regarding product functors, I am fond of the perspective that the “homotopical axiom of unique choice” should be built into the meaning of “there exists a unique”. That is, there is really no difference between asking for a product functor and not asking for one, as long as you define “functor” correctly (in particular, in set-theoretic foundations without AC, the relevant correct definition is anafunctor). The 2-category of categories with non-specified finite products and functors that preserve these is equivalent to the 2-category of categories with functorially specified finite products and functors that preserve those (up to coherent isomorphism, of course).
But other than that, my energy for this discussion has about petered out. (-: I started my comment #10 with “A constructivist would probably say”, and I’ve been trying to take the side of this hypothetical constructivist, but I think I’ve reached the limit of how much I can do that. I do like the argument that objects of categories should be defined to have their structure match as closely as possible the morphisms we are interested in between them.
I still feel, as I said in #17, that taking the filtration but not the attaching maps as part of the structure is bizarre, and I don’t understand conceptually why that’s what one wants, even if in practice it seems to be the useful thing. In particular, it doesn’t seem to generalize very well. If this is the definition of “CW complex”, then how would you define a “cell complex”, where cells don’t have to be attached in order of dimensions and so there is no distinguished filtration?
Regarding product functors, I am fond of the perspective that the “homotopical axiom of unique choice” should be built into the meaning of “there exists a unique”. That is, there is really no difference between asking for a product functor and not asking for one, as long as you define “functor” correctly (in particular, in set-theoretic foundations without AC, the relevant correct definition is anafunctor). The 2-category of categories with non-specified finite products and functors that preserve these is equivalent to the 2-category of categories with functorially specified finite products and functors that preserve those (up to coherent isomorphism, of course).
That sounds appealing. However, I believe that if two definitions of some notion give rise to equivalent categories (or 2-categories or whatever) it doesn’t necessarily mean they’re equally “good” definitions. For example I could define categories with products by specifying loads of junk structure which would then be ignored by morphisms. The 2-category of such things would be equivalent to the usual one, but I suspect you wouldn’t be happy if I submitted such a definition to nLab.
I still feel, as I said in #17, that taking the filtration but not the attaching maps as part of the structure is bizarre, and I don’t understand conceptually why that’s what one wants, even if in practice it seems to be the useful thing. In particular, it doesn’t seem to generalize very well. If this is the definition of “CW complex”, then how would you define a “cell complex”, where cells don’t have to be attached in order of dimensions and so there is no distinguished filtration?
That’s a good question that uncovers more of ad hocery of my proposed definition. I would like to have some conceptual explanation for it, but it really only comes from the premise that cellular maps are the morphisms we want (we could also want all continuous maps, but then I would leave even the filtration out of structure). When it comes to cell complexes there doesn’t seem to be any special kind of morphisms in use besides continuous maps. Also, unlike for CW-complexes, I don’t know any way of using cell complexes which depends on the choice of cells or a filtration. This would lead me to the definition where we have a map with properties without any extra structure. I can’t offer any conceptual way of explaining this or reconciling it with the definition of CW-complexes.
I also feel that this discussion has run its course. I’m just reiterating the same arguments without adding anything new. There’s only the question of what to do with the nLab article on CW-complexes. I wouldn’t have any strong feelings about editing it either way. I made my edit only because I received two thumbs up before this discussion started.
if two definitions of some notion give rise to equivalent categories… it doesn’t necessarily mean they’re equally “good” definitions.
That is undoubtedly true! However, I believe that in this case, the two definitions I cited are “equally good” — or at least that they both have important uses. For instance, it’s often quite useful to know that the 2-category of categories-with-products is 2-monadic over Cat, which is only clear using the definition where they are equipped with product functors.
I can’t offer any conceptual way of explaining this or reconciling it with the definition of CW-complexes.
Maybe that just means CW complexes were a misguided notion to begin with, and cell complexes are more sensible. (-:
I don’t feel the need to change the entry CW complex any more at the moment.
Except that I added a link to this discussion at the bottom. (-:
One of the references looks a little peculiar: is Napier a co-author of Ramachandran, or does Napier refer to something else?
Terrance Napier is coauthor. (fixed). The page numbers were wrong. (fixed) I made a link to the journal… but the journal home page is not very user friendly! :-(
Maybe that just means CW complexes were a misguided notion to begin with, and cell complexes are more sensible. (-:
There may be something to it. It’s true that many things which are done using CW-complexes could be done equally easily (sometimes easier) using cell complexes. But then there are quite a few important things (all related to cellular (co)homology) which I wouldn’t know how to do with general cell complexes. It’s difficult to give any conceptual justification for CW-complexes since they seem to be very specific to the category of spaces while cell complexes make sense pretty much in any category.
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