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I noticed that the entry classifying space is in bad shape. I have added a table of contents and tried to structure it slightly, but much more needs to be done here.
I have added a paragraph on standard classifying spaces for topological principal bundles via the geometric realization of the simplicial space associated to the given topological group.
In the section “For crossed complexes” there is material that had been provided by Ronnie Brown which needs to be harmonized with the existing Idea-section. It proposes something like a general axiomatics on the notion of “classifying space” more than giving details on the geometric realization of crossed complexes
We could possibly use a ‘What does a classifying space classify?’ entry. There are even papers with that as a title. The crossed complex classifying space stuff could go there. (I could see to this if you all like the idea, but not today!)
We could possibly use a ‘What does a classifying space classify?’ entry.
It should be a subsection of that entry.
There are even papers with that as a title.
Yes, I know. i am not sure if that is a particularly useful idea. I think one should consistently speak of geometric realization if that is what one is doing and then ask “What structure is the geometric realization of X a classifying space for?”
what is X here?
$X$ is any object that has a notion of geometric realization.
What I mean is that people tend to speak of “the classifying space of a category” or “the classifying space of a crossed complex” or the like, when they really mean “the geometric realization” of these beasts. That’s why they then have to ask circular-sounding questions like “What does the classifying space of a topological category classify?”
I was suggesting that one should rather say geometric realization if that’s what one means. Because then it makes good sense to ask, for instance “What does the geometric realization of a topological category serve as the classifying space for?”
Urs, that answer seems to pay too much credit to topological spaces. I am not that much of an ’amateur’ of spaces as such, being a bit discrete in my modelling! The basic combinatorics of simplicial classifying things is very simple and it is merely historical that the term classifying space has got attached to them. Taking geometric realisation is not necessarily a good thing. If it is desired for some reason then it has to be done properly.
The point of the question : what does a classifying space classify? is that if we call it a classifying space then it should classify something! Of course, it does, but most sources do not answer the point. For instance, what does the classifying space of a crossed complex classify? If we use the nerve of a crossed complex (without realisation), when probably the best answer would be some sort of fibre bundle or fibration with fibre ’the given crossed complex’, but spelling that out in simple terms is quite awkward. (I mean the general language of the nPOV can give a sense to the idea and a good one, but if someone has a fibration of simplicial sets what do they have to check exactly / explicitly for it to be induced from a map to BC for C a crossed complex. Even with C a crossed module, the structure is difficult to find in the literature. Perhaps that is the wrong question to ask and the only one with sense is the global $(\infty,1)$-language equivalent of it, but taking the geometric realisation seems to obscure the structure from a classical point of view… which is probably where the ’reader’ is coming from, hence my question.)
Tim,
not sure what you are arguing for. Aren’t you actually agreeing with what I said?
Concerning what these things classify: Danny Stevenson and David Roberts have a rather general result that shows that geometric realizations of $\bar W G$ for $G$ a well-pointed topological simplicial group indeed classifies topological $G$-$\infty$-bundles. By restricting to discrete topology, this says what geometric realization of $\bar W G$ for discrete simplicial groups classifies. And since connected crossed complesxes are just another model for certain (strict) $\bar W G$s, that also applies to that case, I’d think.
I was merely reacting to ’geometric realisation’ as being too central. For me $\overline{W}G$ is best called a classifying space since it serves the same task. (I had forgotten about Danny and David’s beautiful result.) Your earlier reply seemed to concentrate on ’space’ whilst my pedagogical point was more on ’classifying’. Somehow ’classifying object’ does not quite seem right (maybe that is what should be used.) So my slight disagreement with what you said is that sticking to the geometric realisation idea (rather than the ’whatever realisation’ idea) puts space in prime position, whilst the various constructions of the classifying ’thingy’ do contain exactly the information that we need to classify something.
I think Urs’ point, which I agree entirely with, is that it doesn’t hardly make sense to call something a “classifying space” (or “classifying blah” for any value of blah) before we know what it classifies! If we have a construction of some thing, whether that thing is a topological space or a simplicial set, out of some input data, then we should name it in some way descriptive of the construction and not call it a “classifying space” until we know what it classifies.
If the thing is a topological space, then “geometric realization” makes sense as a name that describes the construction rather than presuming that it classifies something. If it is a simplicial set, then I suggest that “nerve” is a similarly appropriate word. Homotopically speaking, the operation of geometric realization from simplicial sets to topological spaces is a no-op anyway; it’s the passage from a category to its nerve (where we regard its nerve as a simplicial set, that is as an $\infty$-groupoid) that actually does something (it formally inverts all the morphisms of the category, regarded as an $(\infty,1)$-category, to produce an $\infty$-groupoid).
it formally inverts all the morphisms of the category
at least after passing to a fibrant replacement in the Kan-Quillen model structure, but that is a matter of taste perhaps.
Actually I take that back. Formally inverting all the morphisms of the category (as per Gabriel-Zisman, the general non-fraction version) only gives you the fundamental groupoid of the category, cf Quillen’s Higher algebraic K-theory I. I’m not sure I would call the process of sending (the nerve of) a category to the corresponding Kan complex. One could take the Ex functor as taking a category (qua (oo,1)-category) to an (oo,2)-category, then taking that to an (oo,3)-category and so on. Perhaps one could use (n,r)-categories instead of n=oo, but that is supposition, although it would be nice to take a (1,1)-category to an oo-groupoid by taking n and r to oo together.
@Mike and now I just realised you were thinking of inverting all k-morphisms of the (oo,1)-category associated to a category, but this really needs to be done systematically, as opposed to ’all at once’, because there are a priori no higher morphisms to invert.
I just realised you were thinking of inverting all k-morphisms of the (oo,1)-category associated to a category, but this really needs to be done systematically, as opposed to ’all at once’, because there are a priori no higher morphisms to invert.
Generally, weakly inverting 1-morphisms leads to introducing higher morphisms: the simplicial localization of a category with weak equivalences throws in formal weak inverses with lots of higher morphisms.
Also in the explicit formula for Kan fibrant replacement one can see quite explicitly how the Ex-construciton throws in weak inverse 1-morphisms, 2-morphisms and weak inverse 2-morphisms between these, and so on.
@Urs,
so simplicially localising a category C at the class Mor(C) gives an (oo,0)-category?
Yes. Generally, simplicial localization of a category with weak equivalences gives an $(\infty,1)$-category in which these weak equivalences become actual equivalences.
Heh - you’ve got it Jim. Personally a space for me is always a topological space, unless I’m speaking in the abstract, and mean scheme or manifold or CW complex without dwelling on the details.
By the way, what is the earliest instance of the term ’classifying space’. I note it occurs in Spanier so must be ’early’.
It is not ’space’ that concerns me, rather to explain the ’classifying’ in some way. e.g. a ’classifying simplicial set for principal G-bundles’, a ’classifying spectrum for the cohomology theory blah’ and so on seems fine to me, but just a ’classifying space’ seems a bit meager and vague. The point is ’merely’ pedagogic as I found the current entry on classifying spaces a bit confusing. I agree with Urs that it needs some work on it.
What I mean is that people tend to speak of “the classifying space of a category” or “the classifying space of a crossed complex” or the like, when they really mean “the geometric realization” of these beasts.
I do not fully understand. They mean the geometric realization of the nerve of category. One wants to have a different name for the composition of geometric realization and a nerve from just a geometric realization. Now you propose to call the whole functor just geometric realization ? Even if the word “classifying” is not a priori justified, we use some conditional terminology for the composition. On the other hand, I can imagine moduli space/classifying space constructions which have nothing to do with classical kind of nerves at any stage.
To summarize, I sympathize with not calling a classifying space something what is not yet justified as such (except trivially, every space is a moduli space of its own points), I do not see it easily overcome by calling it geometric realization, for two reasons, first being that it is often actually a composition with a nerve functor of a sort, and second because it does not need to involve nerves and realizations in general to be classifying space or moduli space of something.
David 16
Personally a space for me is always a topological space, unless I’m speaking in the abstract, and mean scheme or manifold or CW complex without dwelling on the details.
In algebraic geometry, algebraic spaces generalize algebraic schemes, and Grothendieck considered more generally that a space is a sheaf of sets of some subcanonical topology on Aff. In any case, in algebraic geometry it is usual, that when you say a space, that you mean something alike scheme, but not as rigid (not in Zariski topology), though on the other hand, typically not a stack. Thus a space is in that context not full generality of “geometry” but certainly the emphasis is, not necessarily a scheme.
Hi Zoran, I also would consider algebraic spaces to be spaces, if I remembered about them in the heat of the moment :)
I have added a bunch of the classical material, statements and proofs, to Classifying space – Examples – For orthogonal and unitary principal bundles.
Was Yoneda aware of the idea of classifying spaces when proving the Yoneda lemma ? (I guess that the Grothendieck, who done it independenty, was aware of).
I think I’ve read that the attribution to Yoneda of this famous lemma is slightly complicated, since he seems not to have actually published a proof himself; see here. In any case, while I don’t know the answer to Zoran’s question, I shouldn’t be surprised if he did know about classifying spaces, since his name is also given to certain topics in homological algebra (I’m thinking for example of $Ext^n(A, B)$ defined by classes of long exact extensions starting with $B$ and ending with $A$), which have close connections to classifying spaces and bar constructions and the like.
I have now added in also all the remaining statements and proofs in the unitary case at Classifying space – Examples – For orthogonal and unitary principal bundles.
What’s an original reference for the construction of classifying spaces (via Grassmannian’s etc)?
The entry currently offers
but I can find no electronic copy of a reference of precisely this title, and in those of similar title I don’t see the definition of classifying spaces.
(Maybe it was me who added that pointer, but I forget.)
What is an(other) record of the original conception of classifying spaces of G-principal bundles?
This opens with
The theory of classifying spaces for principal bundles has a long history in topology [Mi,Se, St]
Maybe for more information:
Thanks! Am adding these…
have added these pointers:
John Milnor, Construction of Universal Bundles, II, Annals of Mathematics Second Series, Vol. 63, No. 3 (May, 1956), pp. 430-436 (jstor:1970012)
Graeme Segal, Classifying spaces and spectral sequences, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)
Norman Steenrod, Milgram’s classifying space of a topological group, Topology Volume 7, Issue 4, November 1968, Pages 349-368 (doi:10.1016/0040-9383(68)90012-8
Jim Stasheff, H-spaces and classifying spaces: foundations and recent developments, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society (1971) pp. 247–272
There is a mention in a Seminar of Moore in Numdam. I have not checked if it does the construction.
Where the References-section started out with “Original articles…” I added “… in topological homotopy theory” and then further down I added “Discussion in simplicial homotopy theory” with all these pointers:
Paul Goerss, Rick Jardine, Section V.3 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996) (doi:10.1007/978-3-0346-0189-4))
David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460)
Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)
Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – Presentations, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 (doi:10.1007/s40062-014-0077-4, arXiv:1207.0249)
added more of the classical textbook references:
Dale Husemoeller, Section 4.12, 4.13 of: Fibre bundles, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. 20, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) (gBooks, pdf)
Norman Steenrod, section II.19 of: The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)
Added pointer to:
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I have added more explicity statement of the Milnor classifying theorem, emphasizing that it works for any Hausdorff structure group and over any paracompact Hausdorff space.
I am wondering about the following, which sounds trivial, but is somewhat subtle:
In which generality are diffeological Cech 1-cocycles relative to a good open cover of a Cartesian space $\mathbb{R}^n$ isomorphic to the trivial cocycle?
The classical Milnor classification theorem implies this for the case that the structure group is a D-topological Hausdorff group. Does it actually fail for non-Hausdorff D-topological groups? How about general diffeological groups?
One should use a method that concretely trivializes the Cech cocycle without arguing indireclt via its classifying space. But constructions of such trivializations that spring to mind turn out to implicitly depend on the fact to be proven.
Linked future pages for BO(n) and BU(n), the classifying spaces of the orthogonal group $O(n)$ and unitary group $U(n)$ respectively. (See discussion on Stiefel-Whitney class.)
I have two questions here: How are formulas like $B O(n)$ and $B U(n)$ supposed to be linked? By removing the formula environment or not? I did the former as otherweise there would have to be an empty space after “B”. I’ve also collected a list of pages, on which the new pages for $B O(n)$, $B S O(n)$, $B U (n)$ and $B S U(n)$ have to be linked. I would do this before creating them. Do I need to write a message for the forum for such a small edit? When having done so for Field, it kinda spammed the forum full with those messages. If it’s necessary, I would spread the edits over a week to avoid this.
Regarding your first question:
the code
[[BO(n)|$B O(n)$]]
works fine for me, I don’t get a spurious whitespace. Do you mean something else?
$\,$
Regarding your second question:
No, minor edits do not need to be announced. I don’t announce many little edits, in order not to flood the nForum (even more than I already do).
When making related edits on a whole range of pages, I usually announce this for only one of them, indicating in the comment which other pages are being edited similarly.
Thank you! I will collect the pages edited to add a link here:
Multiple spaces: Introduction to Cobordism and Complex Oriented Cohomology, Landweber exact functor theorem, Pontrjagin class, Thom’s theorem, basic ideas of moduli stacks of curves and Gromov-Witten theory, classifying space, group cohomology, homology of MG, orthogonal group, special orthogonal group, stable unitary group, string structure, tangential structure, topological K-theory, twisted smooth cohomology in string theory, universal vector bundle
$B O(n)$ only: Thom space, Thom spectrum, real projective space
$B SO(n)$ only: Pontryagin’s theorem, SO(3), SO(4), SO(8), Sullivan model of a spherical fibration
$B U(n)$ only: Atiyah-Segal completion theorem, Chern class, Conner-Floyd Chern class, Pontrjagin ring, Milnor-Quillen theorem on MU, Snaith theorem, Young diagram, algebraic cobordism, basic complex line bundle on the 2-sphere, complex oriented cohomology theory, complex projective space, differential function complex, flux quantization in superconductors – section, formal group, geometry of physics – flux quantization, geometry of physics – fundamental super p-branes, infinite-dimensional sphere, infinity-Chern-Weil theory introduction, line bundle, motivation for sheaves, cohomology and higher stacks, real oriented cohomology theory, unitary group, universal complex line bundle, universal complex orientation on MU, zero-section into Thom space of universal line bundle is weak equivalence
$B SU(n)$ only: SU2-instantons from the correct maths to the traditional physics story, cohomotopy
I intend to publish the pages for BO(n), BSO(n), BU(n), BSU(n) within this week.
promted by discussion in another thread (here) I have reworked the Idea-section of this entry here:
Tried to clarify
the crucial distinction between classifying spaces and moduli stacks (“universes”).
(In particular I shrunk the out-of-place paragraph on the Grothendieck construction to a small side-remark – but this could well be deleted a altogether.)
the fact that there are classifying spaces of other things than principal 1-bundles, for instance of $n$-bundles and of cohomology theories (mentioned EM-spaces as classifying spaces for ordinary cohomology and mentioned the Brown representability theorem). One could expand on this point much more, of course.
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