Lie groupoids as generalized atlases

• Autorzy: Jean Pradines
• Języki publikacji: EN

Abstrakty

EN

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.

Słowa kluczowe

EN

Lie groupoids; spaces of leaves; orbit spaces; 58H05

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2004
• Tom: 2
• Numer: 5
• Strony: 624-662

Daty

wydano

2004-10-01

online

2004-10-01

Twórcy

autor

Jean Pradines

• Université Paul Sabatier

Bibliografia

• [1] [P1] J. Pradines: “Building categories in which a Godement theorem is available”, Cahiers Top. et Géom. Diff., Vol. XVI-3, (1975), pp. 301–306.
• [2] [P2] J. Pradines: “Quotients de groupoïdes différentiables”, C.R.A.S., Paris, Vol. 303(16), (1986), pp. 817–820.
• [3] [P3] J. Pradines: “How to define the graph of a singular foliation”, Cahiers Top. et Géom. Diff., Vol. XVI-4, (1985), pp. 339–380.
• [4] [P4] J. Pradines: “Morphisms between spaces of leaves viewed as fractions”, Cahiers Top. et Géom. Diff., Vol. XXX-3, (1989), pp. 229–246.
• [5] [P5] J. Pradines: “Feuilletages: holonomie et graphes locaux”, C.R.A.S., Paris, Vol. 298(13), (1984), pp. 297–300.
• [6] [P6] J. Pradines: “Remarque sur le groupoïdes cotangent de Weinstein-Dazord”, C.R.A.S., Paris, Vol. 306, (1988), pp. 557–560.
• [7] [B] N. Bourbaki: Variétés différentielles et analytiques, Hermann, Paris, 1971.
• [8] [McL] S. Mac Lane: Categories for the Working Mathematician, Springer-Verlag, New York, 1971.
• [9] [LALG] J.-P. Serre: Lie Algebras and Lie Groups, W.A. Benjamin Inc., New York, 1965.
• [10] [G-Z] P. Gabriel and M. Zisman: Calculus of fractions and homotopy theory, Ergebn. Math. 35, Springer, 1965.
• [11] [Ast. 116] Astérisque 116, Structure transverse des feuilletages, Société Mathématique de France, Paris, 1984. A. Haefliger: “Groupoïdes d'holonomie et classifiants”, pp. 70–97. W.T. van Est: “Rapport sur les S-atlas”, pp. 235–292.

Identyfikatory

DOI

10.2478/BF02475970