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Morphisms between spaces of leaves viewed as fractions. (English) Zbl 0686.57013

This highly categorical paper describes in short (“the world” of smooth groupoids and various types of morphisms (functors). The main ideas arise from Haefliger’s and Van Est’s approach to the transverse geometry of foliations. In particular, the paper contains an abstract re- interpretation of the fundamental group of the space of leaves.
Reviewer: G.Andrzejczak

MSC:

57R30 Foliations in differential topology; geometric theory
18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
58H05 Pseudogroups and differentiable groupoids
57P99 Generalized manifolds
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References:

[1] 1 R. Barre . Quelques propriétés des Q-variétés . Ann. Inst. Fourier. Grenoble 23 ( 1973 ). 227 - 312 . Numdam | MR 348780 | Zbl 0258.57008 · Zbl 0258.57008 · doi:10.5802/aif.478
[2] 2 J. Benabou . Les distributeurs . Univ. Cath. de Louvain. Inst. Math. Pure et Appl. .. Rapport 33 ( 1973 ).
[3] 3 A. Connes . A survey of foliations and operator algebras . I. H. E. S. 1981 . · Zbl 0461.46043
[4] 4 C. Ehresmann . Catégories topologiques et catégories différentiables . Coll. Géom. Diff. Glob. Bruxelles . C.B.R.M. ( 1959 ): reprinted in Charles Ehresmann . Œuvres complètes et commentées. I . Amiens 1983 . 261 - 270 . Zbl 0205.28202 · Zbl 0205.28202
[5] 5 C. Ehresmann . Structures feuilletées . Proc. 5th Canad. Math. Cong. Montréal ( 1961 ): reprinted in Charles Ehresmann . Œuvres complètes et commentées. II-2 . Amiens 1982 . 563 - 626 . Zbl 0146.19501 · Zbl 0146.19501
[6] 6 C. Ehrfsmann . Catégories structurées . Ann. E.N.S. 80 ( 1963 ): reprinted in Charles Ehresmann . Œuvres complètes et commentées. III-1 , Amiens 1980 . Numdam | Zbl 0128.02002 · Zbl 0128.02002
[7] 7 P. Gabriel & M. Zisman . Calculus of fractions and homotopy theory. Ergebn. Math . 35 . Springer 1965 . Zbl 0186.56802 · Zbl 0186.56802
[8] 8 A. Haefliger . Groupoïde d’holonomie et classifiants , Astérisque 116 ( 1984 ), 70 - 97 . MR 755163 | Zbl 0562.57012 · Zbl 0562.57012
[9] 9 M. Hilsum & G. Skandalis , Morphismes K-orientés d’espaces de feuilles et fonctarialité en théorie de Kasporov , Publ. Univ. Paris VI, 75 ( 1985 ); to appear in Ann. E.N.S. Numdam | Zbl 0656.57015 · Zbl 0656.57015
[10] 10 P.T. Johnstone . Topos Theory . Acad. Press 1977 . MR 470019 | Zbl 0368.18001 · Zbl 0368.18001
[11] 11 K. Mackenzie . Lie groupoids and Lie algebroids in Differential Geometry . London Math. Soc. , Lecture Notes Series 124 , London 1987 . MR 896907 | Zbl 0683.53029 · Zbl 0683.53029
[12] 12 S. Mac Lane , Categories for the working mathematician , Springer 1971 . MR 1712872 | Zbl 0906.18001 · Zbl 0906.18001
[13] 13 J. Pradines . Building categories in which a Godement’s Theorem is availabale . Cahiers Top. & Géom. Diff. XVI - 3 ( 1975 ). 301 - 306 . Zbl 0329.18018 · Zbl 0329.18018
[14] 14 J. Pradines , Lois d’action principales conjuguées , Cong. G. M.E.L.. Palma de Mallorca ( 1977 ), communication inédite.
[15] 15 J. Pradines Holonomie et graphes locaux . C. R. A. S. Paris 298 , 13 ( 1984 ). 297 - 300 . MR 765427 | Zbl 0568.57018 · Zbl 0568.57018
[16] 16 J. Pradines , How to define the graph of a singular foliation , Cahiers Top. & Géom. Diff. XVI - 4 ( 1985 ). 339 - 380 . Numdam | MR 816646 | Zbl 0576.57023 · Zbl 0576.57023
[17] 17 J. Pradines , Quotients de groupoïdes différentiables , C. R. A. S. Paris 303 . 16 ( 1986 ). 817 - 820 . MR 872566 | Zbl 0606.57020 · Zbl 0606.57020
[18] 18 J. Pradines & A.A. Alta’ai . Caractérisation universelle du groupe de Haefliger-van Est d’un espace de feuilles ou d’or-bites et Théorème de van Kampen , C. R. A. S. Paris 309 , I ( 1989 ) 503 - 506 . MR 1055468 | Zbl 0698.58053 · Zbl 0698.58053
[19] 19 W.T. Van Est . Rapport sur les S-atlas . Astérisque 116 ( 1984 ), 236 - 292 . MR 755174 | Zbl 0543.58003 · Zbl 0543.58003
[20] 20 H.E. Winkelnkemper . The graph of a foliation . Ann. Glob. Anal. Géom. ( 1982 ). MR 739904 | Zbl 0526.53039 · Zbl 0526.53039 · doi:10.1007/BF02329732
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