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.

]]>added pointer to:

- Loring Tu, Section I.5 in:
*Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies**204**, AMS 2020 (ISBN:9780691191744)

added pointer to:

- Richard Palais, Section 2.6 of:
*The classification of $G$-spaces*, Memoirs of the AMS**36**, 1960 (ISBN:978-0-8218-9979-3 pdf)

[ obsolete ]

]]>added pointer to:

- Michael C. McCord,
*Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (jstor:1995173, pdf)

Added pointer to:

- Gerd Rudolph, Matthias Schmidt, Thm. 3.5.1 of:
*Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 (doi:10.1007/978-94-024-0959-8)

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)

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 a sub-section “References – For equivariant bundles” (here)

]]>There is a mention in a Seminar of Moore in Numdam. I have not checked if it does the construction.

]]>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)90012-8Jim 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

Thanks! Am adding these…

]]>Maybe for more information:

- Stasheff, James D. (1971), “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, pp. 247–272

This opens with

The theory of classifying spaces for principal bundles has a long history in topology [Mi,Se, St]

- [MI] J. Milnor, Construction of universal bundles. II., Ann. of Math. (2)63(1956), 430–436.
- [Se]G. Segal,Classifying spaces and spectral sequences, Inst. Hautes ́Etudes Sci. Publ. Math. No. 34(1968), 105–112.
- [St]N.E. Steenrod,Milgram’s classifying space of a topological group, Topology7(1968), 349–368

What’s an original reference for the construction of classifying spaces (via Grassmannian’s etc)?

The entry currently offers

- Henri Cartan, Laurent Schwartz,
*Le théoréme d’Atiyah-Singer*Séminaire 1963/1964. New York: Benjamin 1967.

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?

]]>Corrected misspelling of Hatcher’s name.

Anonymous

]]>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*.

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.

]]>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 have added a bunch of the classical material, statements and proofs, to *Classifying space – Examples – For orthogonal and unitary principal bundles*.

MR0045432 (13,583b) Chern, Shiing-shen Differential geometry of fiber bundles. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 397–411. Amer. Math. Soc., Providence, R. I., 1952. (Reviewer: H. Samelson), 53.0X

A mini-history of the development of the concept for bundles 9of topological spaces) appears in my

“H-spaces and classifying spaces, I-IV”, AMS Proc. Symp. Pure Math. 22 (1971),247-272. ]]>

Hi Zoran, I also would consider algebraic spaces to be spaces, if I remembered about them in the heat of the moment :)

]]>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.

]]>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.

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