Classifying toposes and foliations
Annales de l'Institut Fourier, Volume 41 (1991) no. 1, pp. 189-209.

For any etale topological groupoid G (for example, the holonomy groupoid of a foliation), it is shown that its classifying topos is homotopy equivalent to its classifying space. As an application, we prove that the fundamental group of Haefliger for the (leaf space of) a foliation agrees with the one introduced by Van Est. We also give a new proof of Segal’s theorem on Haefliger’s classifying space BΓ q .

On démontre que pour tout groupoïde topologique étale G (par exemple, le groupoïde d’holonomie d’un feuilletage) l’espace classifiant est du même type d’homotopie que le topos classifiant. On déduit que le groupe fondamental de l’espace des feuilles au sens de Haefliger est isomorphe à celui de Van Est. En plus, on donne une nouvelle démonstration du théorème de Segal concernant l’espace classifiant BΓ q de Haefliger.

@article{AIF_1991__41_1_189_0,
     author = {Moerdijk, Ieke},
     title = {Classifying toposes and foliations},
     journal = {Annales de l'Institut Fourier},
     pages = {189--209},
     publisher = {Institut Fourier},
     volume = {41},
     number = {1},
     year = {1991},
     doi = {10.5802/aif.1254},
     zbl = {0727.57029},
     mrnumber = {92i:57028},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1254/}
}
TY  - JOUR
AU  - Moerdijk, Ieke
TI  - Classifying toposes and foliations
JO  - Annales de l'Institut Fourier
PY  - 1991
DA  - 1991///
SP  - 189
EP  - 209
VL  - 41
IS  - 1
PB  - Institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1254/
UR  - https://zbmath.org/?q=an%3A0727.57029
UR  - https://www.ams.org/mathscinet-getitem?mr=92i:57028
UR  - https://doi.org/10.5802/aif.1254
DO  - 10.5802/aif.1254
LA  - en
ID  - AIF_1991__41_1_189_0
ER  - 
%0 Journal Article
%A Moerdijk, Ieke
%T Classifying toposes and foliations
%J Annales de l'Institut Fourier
%D 1991
%P 189-209
%V 41
%N 1
%I Institut Fourier
%U https://doi.org/10.5802/aif.1254
%R 10.5802/aif.1254
%G en
%F AIF_1991__41_1_189_0
Moerdijk, Ieke. Classifying toposes and foliations. Annales de l'Institut Fourier, Volume 41 (1991) no. 1, pp. 189-209. doi : 10.5802/aif.1254. http://www.numdam.org/articles/10.5802/aif.1254/

[1] (SGA4) M. Artin, A. Grothendieck, J.-L. Verdier, Théorie de topos et cohomologie des schémas, SLN, 269, 270 (1972). | Zbl

[2] M. Artin, B. Mazur, Etale homotopy, SLN, 100 (1969). | MR | Zbl

[3] R. Barre, De quelques aspects de la théorie des Q-variétés différentielles et analytiques, Ann. Inst. Fourier, Grenoble, 23-3 (1973), 227-312. | EuDML | Numdam | MR | Zbl

[4] R. Bott, Characteristic classes and foliations, in : Lectures on Algebraic and Differential Topology, SLN, 279 (1972), 1-94. | Zbl

[5] M. Bunge, An application of descent to a classification theorem for toposes, McGill University, preprint, 1988. | Zbl

[6] P. Deligne, Théorie de Hodge III, Publ. Math. IHES, 44 (1975), 5-77. | EuDML | Numdam | Zbl

[7] R. Diaconescu, Change of base for toposes with generators, J. Pure and Appl. Alg., 6 (1975), 191-218. | MR | Zbl

[8] J. Duskin, Simplicial methods and the interpretation of “triple” cohomology, Memoirs AMS, 163 (1975). | MR | Zbl

[9] W.T. Van Est, Rapport sur les S-atlas, Astérisque, 116 (1984), 235-292. | Zbl

[10] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory, Springer-Verlag, 1967. | Zbl

[11] J. Giraud, Classifying topos ; in : F.W. Lawvere (ed.), Toposes, Algebraic Geometry and Logic, SLN, 274 (1972), 43-56. | MR | Zbl

[12] A. Grothendieck, Revêtements étales et groupe fondamental, SLN, 224 (1971) (SGA I). | Zbl

[13] A. Haefliger, Feuilletages sur les variétés ouvertes, Topology, 1 (1970), 183-194. | MR | Zbl

[14] A. Haefliger, Homotopy and Integrability, in : Manifolds, Amsterdam 1970, SLN, 197 (1971), 133-163. | MR | Zbl

[15] A. Haefliger, Groupoïde d'holonomie et classifiants, Astérisque, 116 (1984), 235-292. | Zbl

[16] L. Illusie, Complexe cotangent et déformations II, SLN, 283 (1972). | MR | Zbl

[17] J.F. Jardine, Simplicial objects in a Grothendieck topos, in : Contemporary Mathematics, vol. 55, part I (1986), 193-239. | MR | Zbl

[18] J.F. Jardine, Simplicial presheaves, J. Pure and Appl. Alg., 47, 35-87. | MR | Zbl

[19] P.T. Johnstone, Topos Theory, Academic Press, 1977. | MR | Zbl

[20] A. Joyal, Letter to A. Grothendieck, (1984).

[21] A. Joyal, G. Wraith, K(π, n)-toposes, in : Contempary Mathematics, vol 30 (1983).

[22] A. Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Memoirs AMS, 309 (1984). | MR | Zbl

[23] I. Moerdijk, The classifying topos of a continous groupoid I, Transactions AMS, 310 (1988), 629-668. | MR | Zbl

[24] I. Moerdijk, Toposes and Groupoids, in : F. Borceux (ed), Categorical Algebra and its Applications, SLN, 1348 (1988), 280-298. | MR | Zbl

[25] P. Molino, Sur la géométrie transverse des feuilletages, Ann. Inst. Fourier, Grenoble, 25-2 (1975), 279-284. | Numdam | MR | Zbl

[26] J. Pradines, A.A. Alta'Ai, Caractérisation universelle du groupe de van Est d'une espace de feuilles ou d'orbites, et théorème de van Kampen, preprint, 1989. | Zbl

[27] J. Pradines, J. Wouafa-Kamga, La catégorie des QF-variétés, CRAS (série A), 288 (1979), 717-719. | MR | Zbl

[28] D. Quillen, Higher Algebraic K-theory : I ; in Springer LNM, 341 (1972), 85-147. | MR | Zbl

[29] G.B. Segal, Classifying spaces and spectral sequences, Publ. Math. IHES, 34 (1968), 105-112. | Numdam | MR | Zbl

[30] G.B. Segal, Categories and cohomology theories, Topology, 13 (1974), 304-307. | MR | Zbl

[31] G.B. Segal, Classifying spaces related to foliations, Topology, 17 (1978), 367-382. | MR | Zbl

[32] B. Saint-Donat, Techniques de descente cohomologique, in SLN 270 (cf. [1] above), 83-162. | Zbl

[33] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Ac. Sc., 42 (1956), 359-363. | MR | Zbl

[34] J. Tapia, Sur la cohomologie de certains espaces d'orbites, thèse, Univ. Paul Sabatier, Toulouse, 1987.

Cited by Sources: