Journal of the Mathematical Society of Japan

The category of reduced orbifolds in local charts

Anke D. POHL

Full-text: Open access


It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective étale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category.

Article information

J. Math. Soc. Japan, Volume 69, Number 2 (2017), 755-800.

First available in Project Euclid: 20 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R18: Topology and geometry of orbifolds 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05]
Secondary: 58H05: Pseudogroups and differentiable groupoids [See also 22A22, 22E65]

reduced orbifolds orbifold maps groupoids groupoid homomorphisms


POHL, Anke D. The category of reduced orbifolds in local charts. J. Math. Soc. Japan 69 (2017), no. 2, 755--800. doi:10.2969/jmsj/06920755.

Export citation


  • A. Adem, J. Leida and Y. Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007.
  • J. Borzellino and V. Brunsden, A manifold structure for the group of orbifold diffeomorphisms of a smooth orbifold, J. Lie Theory, 18 (2008), 979–1007.
  • W. Chen and Y. Ruan, Orbifold Gromov–Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85.
  • A. Haefliger, Groupoï des d'holonomie et classifiants, Astérisque, No. 116, (1984), 70–97.
  • E. Lerman, Orbifolds as stacks?, Enseign. Math. (2), 56 (2010), 315–363.
  • E. Lupercio and B. Uribe, Gerbes over orbifolds and twisted $K$-theory, Comm. Math. Phys., 245 (2004), 449–489.
  • I. Moerdijk, Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205–222.
  • I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, 91, Cambridge University Press, Cambridge, 2003.
  • I. Moerdijk and D. Pronk, Orbifolds, sheaves and groupoids, K-Theory, 12 (1997), 3–21.
  • D. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math., 102 (1996), 243–303.
  • I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A., 42 (1956), 359–363.
  • I. Satake, The Gauss–Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan, 9 (1957), 464–492.
  • A. Schmeding, The diffeomorphism group of a non-compact orbifold., Diss. Math., 507 (2015), 179.
  • W. Thurston, The geometry and topology of three-manifolds, available at