Abstract
We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. \(\mathbb {P}_n\)-algebras. Roughly speaking, a coisotropic morphism is given by a \(\mathbb {P}_{n+1}\)-algebra acting on a \(\mathbb {P}_n\)-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.
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Acknowledgements
We would like to thank D. Calaque, G. Ginot, M. Porta, B. Toën and G. Vezzosi for many interesting and stimulating discussions. The work of P.S. was supported by the EPSRC grant EP/I033343/1. We also thank the anonymous referee for his/her useful comments.
