Derived coisotropic structures I: affine case

Abstract

We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a $P_n$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.

49 pages. v2: many proofs rewritten and the paper is split into two parts

References in corpus (5)

• Derived coisotropic structures II: stacks and quantization
• Braces and Poisson additivity
• Calabi-Yau structures on topological Fukaya categories
• Quantisation of derived Lagrangians
• Maurer-Cartan spaces of filtered L-infinity algebras

Cited by in corpus (17)

• Shifted cotangent stacks are shifted symplectic
• Derived coisotropic structures II: stacks and quantization
• Braces and Poisson additivity
• Shifted Coisotropic Correspondences
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• Quantisation of derived Lagrangians
• Degenerate Classical Field Theories and Boundary Theories
• Structures symplectiques et de Poisson sur les champs en catégories
• Shifted Poisson structures on differentiable stacks
• On Quantum Obstruction Spaces and Higher Codimension Gauge Theories
• Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks
• Shifted coisotropic structures for differentiable stacks
• A note on étale atlases for Artin stacks and Lie groupoids, Poisson structures and quantisation
• Shifted Poisson structures and moduli spaces of complexes
• Gaiotto's Lagrangian subvarieties via derived symplectic geometry
• Exact triangles, Koszul duality, and coisotopic boundary conditions
• Deformation Quantization via Categorical Factorization Homology