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Homotopy types of topological stacks · Around

2008

Homotopy types of topological stacks

Noohi, Behrang

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We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence.

Under a certain paracompactness condition on \X, we show that X is actually well-defined up to homotopy equivalence.
These results are formulated in terms of functors from the category of topological stacks to the (weak) homotopy category of topological spaces.
We prove similar results for (small) diagrams of topological stacks.

Built on

A. Dold, Partitions of unity in the theory of fibrations , Ann. of Math. (2) 78
1963
Earlier work this paper cites.
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory , Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag (1967)
1967
Earlier work this paper cites.

Similar

A. Haefliger, Homotopy and integrability , 1971 Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, 133–163
1970
Cited alongside, same era.
G. W. Whitehead, Elements of Homotopy Theory , Graduate Texts in Mathematics, 61. Springer-Verlag, New York–Berlin, 1978
1978
Cited alongside, same era.
K. Behrend, G. Ginot, B. Noohi, and P. Xu, String topology for stacks , arXiv:0712.3857v1 [math.AT]
Original
Cited in the paper.
J. Ebert, J. Giansiracusa, Pontrjagin-Thom maps and the homology of the moduli stack of stable curves , to appear in Math. Annalen
Cited in the paper.
B. Noohi, Foundations of topological stacks, I , math.AG/0503247v1
Cited in the paper.
B. Noohi, Mapping stacks of topological stacks , to appear in Crelle
Cited in the paper.

Then

S. Mac Lane, Categories for the Working Mathematician , Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998
1998
Later among the works it cites.
K. Behrend, Cohomology of stacks , Intersection theory and moduli, 249-294 (electronic) ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004
2004
Later among the works it cites.

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