Deformation theory and cotangent complex of dg operads

Abstract

In the first part, we give an explicit description of the cotangent complex of differential graded (dg) operads, modeled as an operadic infinitesimal bimodule. This leads to a uniform formula for the Quillen cohomology of their associated algebras. We further show that the cotangent complex of the dg $E_\infty$-operad is represented by the Pirashvili functor, while that of the dg $E_n$-operad is conveniently described via its Hochschild complex. In the second part, we establish an explicit relation between deformation theory and (spectral) Quillen cohomology for various types of algebraic objects. Combining these results, we obtain a formulation of the space of first-order deformations of dg operads, which is particularly convenient in the case of dg $E_n$-operads.