Deformation theory and cotangent complex of dg operads

TL;DR

This paper develops a detailed framework for understanding the deformation theory and cotangent complexes of dg operads, providing explicit formulas and relations to Quillen cohomology, especially for dg $E_ $-operads.

Contribution

It offers explicit descriptions of cotangent complexes for dg operads and links deformation theory with Quillen cohomology, advancing the understanding of algebraic deformations.

Findings

Cotangent complex of dg operads described explicitly.

Representation of cotangent complex of dg $E_$-operad by Pirashvili functor.

Relation between deformation theory and spectral Quillen cohomology established.

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.