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Structure of mapping objects in the category of orbifolds Pronk, Dorette


We consider topological orbifolds as proper \'etale groupoids, i.e., topological groupoids with a proper diagonal and \'etale structure maps. We call these orbigroupoids. To describe maps between these groupoids and 2-cells between them, we will use the bicategory of fractions of the 2-category of orbigroupoids and continuous functors with respect to a subclass of the Morita equivalences which is gives a bicategory of fractions that is equivalent to the usual one and renders mapping groupoids that are small. We will present several nice results about the equivalence relation on the 2-cell diagrams in this bicategory that then enable us to obtain a very explicit description of the topological groupoid $\mbox{Map}\,(G,H)$ encoding the new generalized maps from $G$ to $H$ and equivalence classes of 2-cell diagrams between them, for any orbigroupoids $G$ and $H$. In particular, we obtain the arrow space as a retract of the space of all 2-cell diagrams. When $G$ has a compact orbit space we show that the mapping groupoid is an orbigroupoid and has the appropriate universal properties to be the mapping object. In particular, sheaves on this groupoid for the mapping topos for geometric morphisms between the toposes of sheaves on $G$ and $H$. This groupoid can also be viewed as a pseudo colimit of mapping groupoids in the original 2-category of topological groupoids and continuous functors.

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