Derived coisotropic structures I: affine case

TL;DR

This paper introduces a new framework for understanding coisotropic structures in shifted Poisson geometry, linking them to Maurer--Cartan elements and constructing operadic replacements, advancing the theoretical foundation of derived algebraic geometry.

Contribution

It defines coisotropic structures on morphisms of commutative dg algebras, identifies their space with Maurer--Cartan elements, and constructs a cofibrant operadic replacement, providing new tools for shifted Poisson geometry.

Findings

Identification of coisotropic structures with Maurer--Cartan elements

Construction of a cofibrant operadic replacement

Equivalence of morphisms of shifted Poisson algebras with coisotropic structures on graphs

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.