A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation (Q6596135)

Artin stacks were introduced by \textit{M. Artin} in [Invent. Math. 27, 165--189 (1974; Zbl 0317.14001)]. When attempting to formulate Poisson structures or deformation quantization for Artin stacks, derived Artin stacks or Lie groupoids, one immediately encounters the difficulty that such structures on affine schemes are only functorial with respect to étale morphisms (local diffeomorphisms or biholomorphisms in differrentiable and analytic settings). The main result of this paper (Theorem 2.1) claims that when \(\mathfrak{X}\)\ is an Artin stack, \(D_{\ast}\mathfrak{X}\)\ admits an étale atlas by stacky affines. The theorem has two dividends.\N\N\begin{itemize}\N\item[(1)] The formulation of concepts such as Poisson structures and deformation quantizations for Artin stacks and Lie groupoids.\N\N\item[(2)] It provides an environment where we can compare \textsf{NQ}-manifolds and Artin stacks directly.\N\end{itemize}\N\NThere are generalizations to derived Artin stacks using stacky derived affines, whose rings of functions are the stacky CDCAs (Commutative Differential Graded Algebras) of [\textit{J. P. Pridham}, J. Topol. 10, No. 1, 178--210 (2017; Zbl 1401.14017), \S 3.1], given by introducing a second grading and a chain algebra structure. It is shown (Theorem 4.12) that, for a derived Artin \(n\)-stack, the associated functor \(D_{\ast}\mathfrak{X}\)\ on stacky derived affines admits an étale atlas. There are also analogues for (derived) differentiable stacks, in which setting stacky affines correspond to \textsf{NQ}-manifolds, and for (derived) analytic stacks in both Archimedian and non-Archimedian contexts.