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

Abstract

We explain how any Artin stack $X$ over $Q$ 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 $X$. This site has the same quasi-coherent sheaves as $X$ and leads to efficient formulations of shifted Poisson structures, differential operators and deformation 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.