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

Abstract

We explain how any Artin stack X over Q extends to a functor on nonnegatively 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 quantisations for Artin stacks. There are generalisations to higher and derived stacks, and analogues for differentiable and analytic stacks. This note is just a slight elaboration of constructions scattered across several of the author's papers. J.P.PRIDHAM 4.10). There are also analogues for (derived) differentiable stacks, in which setting stacky affines correspond to NQ-manifolds, and for (derived) analytic stacks in both Archimedean and non-Archimedean contexts ( §4.2).

Shifted cotangent stacks are shifted symplectic

“…Moreover, the latter notion is formulated in terms of hypergroupoids of smooth manifolds, while we consider ordinary groupoids, but allow the individual spaces to be derived stacks. We refer to [Pri17;Pri18a;Pri19b] for a description of shifted symplectic structures on derived Artin stacks presented in terms of hypergroupoids in algebraic and differential geometry.…”