2013/04/25 by Matteo Tommasini, Tommasini, Matteo
Mathematics · Physics and Astronomy · #14D20 #14F45 #14H51 #14H60 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Category Theory (math.CT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1304.6959
openalex publication_date 2013/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe a bicategory (Red Orb) of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we first define a 2-category (Red Atl) whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definitions of 2-morphisms and compositions of them is new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct the bicategory (Red Orb) from the 2-category (Red Atl). We prove that (Red Orb) is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, étale Lie groupoids by D. Pronk and I. Moerdijk and to the 2-category of reduced orbifolds described by several authors in the past in terms of a suitable class of differentiable Deligne-Mumford stacks.