Bicategories of action groupoids (Q6561384)

This paper shows that the 2-category of action LIe groupoids localized in the following three distinct ways yield equivalent bicategories, which are generalizations of the known case of representable oribifold groupoids [\textit{D. Pronk} and \textit{L. Scull}, Can. J. Math. 62, No. 3, 614--645 (2010; Zbl 1197.57026); Can. J. Math. 69, No. 4, 851--853 (2017; Zbl 1421.57046)]:\N\N\begin{itemize}\N\item[(1)] Localizing at equivariant weak equivalences à la \textit{D. A. Pronk} [Compos. Math. 102, No. 3, 243--303 (1996; Zbl 0871.18003)];\N\N\item[(2)] Localizing by use of surjective submersive equivariant weak equivalences and anafunctors à la \textit{D. M. Roberts} [Categ. Gen. Algebr. Struct. Appl. 15, No. 1, 183--229 (2021; Zbl 1502.18016); Theory Appl. Categ. 26, 788--829 (2012; Zbl 1275.18023)];\N\N\item[(3)] Localizing at all weak equivalences.\N\end{itemize}\N\NIt is also shown that the decomposition of equivariant weak equivalences used in [\textit{D. Pronk} and \textit{L. Scull}, Can. J. Math. 62, No. 3, 614--645 (2010; Zbl 1197.57026), Proposition 3.5] also applies in this more general setting, which allows of breaking down equivariant weak equivalences into two specific types, namely, projections and inclusions.