Derived derivations govern contraderived deformations of dg algebras over dg (pr)operads
J. P. Pridham
TL;DR
The paper resolves when derived deformations of a dg algebra over a dg (pr)operad can be governed by a DGLA by replacing derived deformations with contraderived deformations in Positselski’s framework. It develops a comprehensive operadic approach, using the hat/twist constructions to encode deformations as maps of coloured dg (pr)operads and showing that the space of contraderived deformations is controlled by the augmented deformation complex $\underline{\mathrm{Der}}_{\bar{\Omega}\mathcal{C}}(\gamma)$. A key result is that, for cofibrant algebras, the mapping-space perspective via contraderived categories is equivalent to the Maurer–Cartan theory of the corresponding derivation DGLA, and this equivalence extends to dg properads. The framework provides a universal deformation theory: among well-behaved functors with deformation theory, the contraderived dg category best approximates the derived category, and Hinich’s counterexamples are resolved within this contraderived viewpoint. These findings unify derived and contraderived perspectives, clarifying when and how a DGLA governs deformations across operadic and properadic contexts.
Abstract
We show that Hinich's simplicial nerve of the differential graded Lie algebra (DGLA) of derived derivations of a dg algebra $A$ over a dg properad $\mathcal{P}$ is equivalent to the space of deformations of $A$ as a $\mathcal{P}_{\infty}$-algebra in Positselski's contraderived dg category. This resolves Hinich's counterexamples to the general existence of derived deformations. It also generalises his results when $A$ is homologically bounded below, since contraderived deformations are then precisely derived deformations.