Orbifolds as groupoids: an introduction

By Ieke Moerdijk. 2002

246 citations recorded by the source corpus.

This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ec0580a015ab4a30605605c05b1661b9df5808ea

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In Mathematical Physics

Citations in the corpus

Integration theory on the zero sets of polyfold Fredholm sections (2007, unknown): “in [21] and [22].”
On an extension of the notion of Reedy category (2008, background): “Such complexes of groups can be used to model orbifold structures on a triangulated space |X |, see [21,29].”
Asymptotic expansion of the heat kernel for orbifolds (2008, background): “mannian structure and constructing the “bundle” of orthonormal frames. The orthonormal frame bundle is actually an orbibundle, the appropriate notion of vector bundle over an orbifold as described in [19] or [30]. However, its total space is a smooth manifold with an action of the orthogonal group, ”
Sc-Smoothness, Retractions and New Models for Smooth Spaces (2010, unknown): “A polyfold is a generalization of the modern notion of orbifold as presented in [16].”
Differentiable stacks and gerbes (2006, background): “, étale proper groupoids, a viewpoint largely due to Haefliger, is quite useful [35].”
The orbifold Chow ring of toric Deligne-Mumford stacks (2003, methodology): “To describe the connection between the combinatorics of the stacky fan � and the substacks of X(�), we use the theory of groupoids; see [ 20 ] for an introduction.”
Presentations of noneffective orbifolds (2003, background): “map ψP,x,y by conjugating in the target. If P is an equivalence, as above, the construction works in both directions to give inverse isomorphisms. 2.4. Orbifold Groupoids. Moerdijk and Pronk ([MP97], [Moe]) show that to an eﬀective orbifold with a given atlas, one can naturally associate a smooth pr”
Lectures on integrability of Lie brackets (2006, unknown): “Other fundamental examples were discovered over many years by different authors, such as Élie Cartan’s infinite continuous groups (Singer and Sternberg, [56]), the integrability of infinitesimal actions on manifolds (Palais, [50], Moerdijk and Mrčun [47]), abstract Atiyah sequences and transitive ”
Stacks of quantization-deformation modules on complex symplectic manifolds (2003, background): “Lie groupoids or better, differentiable stacks (see [ 15 ], [4]).”
The Euler characteristic of a category (2006, background): “On the other hand, the groups G(p) fit together to form a complex of finite groups on P , that is, a weak functor G : P op ✲ Cat taking values in finite groups (regarded as one-object categories) and injective homomorphisms; see §3 of [M].”
A General Fredholm Theory and Applications (2005, background): “generalization. The approach via groupoids seems particularly useful, even from an analysis viewpoint. We refer the reader to the excellent article by Moerdijk about a groupoid approach to orbifolds, [31], as well as the book [32]. 1.3.1. Polyfold groupoids. Recall that a groupoid Gis a small catego”
A general Fredholm theory III: Fredholm functors and polyfolds (2008, methodology): “Here again we have an underlying topological space equipped with an the overhead of a Morita-equivalence class of étale proper Lie groupoids as described in [43].”
A general Fredholm theory I: A splicing-based differential geometry (2006, background): “The third paper, [9], develops the Fredholm theory in polyfolds, which could be viewed as a theory of Fredholm functors in a version of Lie groupoids with object and morphism spaces build on the new local models (see [17], [18] for the groupoid concepts in a manifold world).”
Representation and character theory in 2-categories (2006, background): “[Moe02], is a Lie groupoid G such that all stabilizer groups HomΓ(x, x), x ∈ Ob(Γ), are finite.”