Deformation theory and cotangent complex of dg operads

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.

💡 Research Summary

This paper is divided into two main parts, each addressing a different but closely related aspect of differential graded (dg) operads: the explicit description of their cotangent complexes and the connection between deformation theory and Quillen cohomology.

In the first part the authors work with a Σ‑cofibrant, connective dg operad (P) over a commutative ring (k). Using the recent framework of Harpaz‑Nuiten‑Prasma on operadic tangent categories, they establish a Quillen equivalence between the tangent category (T_{P}\operatorname{Op}(C(k))) and the category (\operatorname{IbMod}(P)) of infinitesimal (P)‑bimodules. Under this equivalence the cotangent complex (L_{P}) is identified with a very concrete infinitesimal bimodule: for each colour tuple ((c_{1},\dots,c_{m};c)) the component is the (m)-fold direct sum of the underlying operation complex (P(c_{1},\dots,c_{m};c)). Equivalently one can write (L_{P}\cong P\circ (1),I_{C}), the infinitesimal composite of (P) with the initial coloured operad.

This explicit model yields a uniform formula for the Quillen cohomology of any (P)‑algebra (A) with coefficients in an (A)‑module (N). Instead of the classical description via the module category (\operatorname{Mod}{A}^{P}) and the Kähler differentials (\Omega{A}), the authors show that \