关于 Artin 栈与 Lie 群胚的 étale 图册、Poisson 结构与量子化的注记
A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation
AI总结：
本文解释了 Artin 栈扩张为非负分次交换上链代数函子的方法，利用 stacky 仿射概形的 étale 图册构建小 étale 景，从而高效表述偏移 Poisson 结构与量子化，并将类似构造推广至 Lie 群胚与可微及解析栈。
摘要：
我们解释了定义在 $\mathbb{Q}$ 上的任何 Artin 栈 $\mathfrak{X}$ 如何扩张为非负分次交换上链代数上的函子，我们将其视为 Lie 代数胚或 stacky 仿射概形上的函数。对于这些 CDGA 存在 étale 态射的概念，并且 Artin 栈允许通过 stacky 仿射概形构成 étale 图册，从而在 $\mathfrak{X}$ 上生成由 stacky 仿射概形构成的小 étale 景。该景具有与 $\mathfrak{X}$ 相同的拟凝聚层，并促成了 Artin 栈上偏移 Poisson 结构、微分算子与量子化的高效表述。这些结论可推广至更高栈与导出栈。我们还描述了可微栈与解析栈的类似物；特别地，Lie 群胚自然给出 NQ-流形上的函子，借此我们可以转移结构。在这些背景下，局部微分同胚与双全纯映射是 étale 态射的类似物。本注记主要阐述了散见于作者数篇论文中的构造，但着重强调了点函子视角。新结果包括一致性检验，表明在 stacky 仿射概形上诱导的向量丛或挠子等结构的概念，与基于平坦联络的熟悉定义相一致。
英文摘要：
We explain how any Artin stack $\mathfrak{X}$ over $\mathbb{Q}$ extends to a functor on non-negatively graded commutative cochain algebras, which we think of as functions on Lie algebroids or stacky affine schemes. There is a notion of étale morphisms for these CDGAs, and Artin stacks admit étale atlases by stacky affines, giving rise to a small étale site of stacky affines over $\mathfrak{X}$. This site has the same quasi-coherent sheaves as $\mathfrak{X}$ and leads to efficient formulations of shifted Poisson structures, differential operators and quantisations for Artin stacks. There are generalisations to higher and derived stacks. We also describe analogues for differentiable and analytic stacks; in particular, a Lie groupoid naturally gives a functor on NQ-manifolds which we can use to transfer structures. In those settings, local diffeomorphisms and biholomorphisms are the analogues of étale morphisms. This note mostly elaborates constructions scattered across several of the author's papers, but with an emphasis on the functor of points perspective. New results include consistency checks showing that the induced notions of structures such as vector bundles or torsors on a stacky affine scheme coincide with familiar definitions in terms of flat connections.
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