In the theory of differential graded (co)operads, the notion of twisting is ubiquitous. The fundamental notion is the twisting map from a cooperad $C$ to an operad $P$. It is defined as a Maurer-Cartan element in the dg Lie algebra $\operatorname{Hom}(C,P)$.
Following Loday and Vallette's "Algebraic operads", bar-cobar duality is proven via studying the twisted composite $P \circ_\pi BP$ which is constructed from the twisting map $\pi:BP \rightarrow P$ which exists since $BP$ is the cofree cooperad on $P$. It is shown the acyclicity of this complex is equivalent to $\Omega B P \simeq P$.
More classically, Maurer-Cartan elements in a dg Lie algebra can be used to define the twisting of a dg Lie algebra. For example, if we use Quillen's equivalence of rational dg Lie algebras and rational spaces, twisting by a Maurer-Cartan element corresponds to changing basepoints, so a useful analogy is that the Maurer-Cartan elements up to homotopy are like the path components of your Lie algebra.
Further twisting techniques were introduced by Willwacher when studying Kontsevich's graph complex. These apply when one receives a map from the Lie operad and allow one to talk about twisting elements of algebras over non-Lie operads. This can be used to streamline the proof of the formality of $E_n$.
Now it is known by work of Ching that bar-cobar duality holds in the setting of spectra, but through very different techniques. Namely, there seems to be no map $BP \rightarrow P$ in the land of spectra, so Ching is forced to directly prove there are zigzags of equivalences from $\Omega B P$ to $P$. Ching, in another paper, also proves the Koszul dual of the commutative operad deserves to be called the spectral Lie operad.
So here are some basic questions:
- Are there notions of Maurer-Cartan elements of a spectral Lie algebra?
- Is there a notion of twisted composite product of operads and cooperads in spectra?
- Is there a notion of operadic twisting in spectra?
