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**The category of reduced orbifolds in local charts.**
*(English)*
Zbl 1381.57016

This article is a welcome contribution to the understanding of smooth orbifolds as it provides a missing bridge between the traditional approach to orbifolds (see I. Satake [Proc. Natl. Acad. Sci. USA 42, 359–363 (1956; Zbl 0074.18103)] and W. P. Thurston [Three-dimensional geometry and topology. Princeton, NJ: Princeton University Press (1997; Zbl 0873.57001)]) using local charts, and the groupoid approach introduced by I. Moerdijk and D. A. Pronk [\(K\)-Theory 12, No. 1, 3–21 (1997; Zbl 0883.22005)]. Even though the article of Moerdijk and Pronk established that smooth orbifolds defined by local charts correspond to Morita equivalence classes of proper étale Lie groupoids, this equivalence is not an isomorphism. This insufficiency makes a groupoid approach to questions involving the differential topology of orbifolds difficult and, further, has resulted in a proliferation of definitions and ad-hoc techniques each tailored to overcome a particular challenge.

The author approaches this problem by defining a category in terms of marked proper effective étale Lie groupoids which is isomorphic to the orbifold category. In particular, the author first recalls the construction of a groupoid from an orbifold atlas representing the orbifold structure of an orbifold. These are referred to as atlas groupoids. These atlas groupoids are then given a marking. There is a natural notion of homomorphism between between marked atlas groupoids which can be characterized in local charts. On the orbifold side, this leads to a notion of charted orbifold map. An orbifold map is then defined as a certain equivalence class of charted orbifold maps, the point being that such maps are equivalent if and only if they induce the same charted orbifold map on common refinements of the orbifold atlases. This leads to a category of reduced orbifolds. Next, orbifolds are characterized as equivalence classes of marked atlas groupoids and orbifold maps as equivalence classes of homomorphisms of these groupoids. The result is an isomorphism between the category of marked atlas groupoids and the category of orbifolds. The final step is to show that the category of marked atlas groupoids is isomorphic to a category of marked proper effective étale Lie groupoids.

The author approaches this problem by defining a category in terms of marked proper effective étale Lie groupoids which is isomorphic to the orbifold category. In particular, the author first recalls the construction of a groupoid from an orbifold atlas representing the orbifold structure of an orbifold. These are referred to as atlas groupoids. These atlas groupoids are then given a marking. There is a natural notion of homomorphism between between marked atlas groupoids which can be characterized in local charts. On the orbifold side, this leads to a notion of charted orbifold map. An orbifold map is then defined as a certain equivalence class of charted orbifold maps, the point being that such maps are equivalent if and only if they induce the same charted orbifold map on common refinements of the orbifold atlases. This leads to a category of reduced orbifolds. Next, orbifolds are characterized as equivalence classes of marked atlas groupoids and orbifold maps as equivalence classes of homomorphisms of these groupoids. The result is an isomorphism between the category of marked atlas groupoids and the category of orbifolds. The final step is to show that the category of marked atlas groupoids is isomorphic to a category of marked proper effective étale Lie groupoids.

Reviewer: Joseph E. Borzellino (San Luis Obispo)