On orbifold embeddings (Q2868512)

In this article, the authors study the notion of suborbifold, and more generally, orbifold embeddings using the groupoid approach to orbifolds. The motivation for this study was the authors' realization that the classical understanding of suborbifold as defined originally by \textit{W. Thurston} [``The geometry and topology of 3--manifolds,'' Lecture Notes, Princeton University Mathematics Department (1978)], was inadequate for a proper study of the Fukaya category of symplectic orbifolds. In particular, the authors note the diagonal embedding \(\mathcal{O}\to\mathcal{O}\times\mathcal{O}\) of an orbifold into its product orbifold was not a suborbifold of the product in this restricted sense. This difficulty with the diagonal embedding had been observed before in work of the reviewer and \textit{V. Brunsden} [in: Banyaga, Augustin (ed.) et al., Infinite dimensional Lie groups in geometry and representation theory. Papers delivered on the occasion of the 2000 Howard fest on infinite dimensional Lie groups in geometry and representation theory, Washington, DC, USA, August 17--21, 2000. Singapore: World Scientific. 116-137 (2002; Zbl 1042.57013)] and [J. Lie Theory 18, No. 4, 979--1007 (2008; Zbl 1166.57021)], where a suitable notion of suborbifold is defined based on the classical ``atlas of charts'' approach to orbifolds. The authors of this article first review the groupoid approach to orbifolds and define their notion of orbifold embedding (which is a minor modification of the notion that appears in [\textit{A. Adem, J. Leida} and \textit{Y. Ruan}, Orbifolds and stringy topology. Cambridge Tracts in Mathematics 171. Cambridge: Cambridge University Press. (2007; Zbl 1157.57001)]. They discuss properties of orbifold embeddings, present many examples, and prove that given an abelian orbifold embedding, the induced map between inertia orbifolds is an orbifold embedding as well. Also, since the notion of orbifold embedding defined here is not Morita invariant, the last sections of the paper seek to address this issue for the special case of translation groupoids.