An important example in homotopy theory is given by operadic twisting. This was introduced by T. Willwacher in "M. Kontsevich’s graph complex and the Grothendieck–Teichmüller Lie algebra" (Inventiones Math. 200 (2015), 671–760), and its comonadicity was proved by V. Dolgushev and T. Willwacher in Operadic twisting – With an application to Deligne's conjecture (J. Pure Appl. Alg. 219 (2015), Issue 5, 1349-1428).

Essentially, for a dg operad $\mathcal{O}$ equipped with a morphism from the operad $\mathrm{Lie}$ (or $\mathrm{Lie}_\infty$), one can construct a new operad $\mathrm{Tw}(\mathcal{O})$. It is done in two steps. First, one considers the operad $\mathrm{MC}(\mathcal{O})$ whose algebras are $\mathcal{O}$-algebras with a given solution to the Maurer--Cartan equation $$d\alpha+\frac12[\alpha,\alpha]=0.$$ Then one "twists" the differential $d_{\mathrm{MC}}$ of that latter operad by adding the "commutator with the commutator" (the operadic commutator with the unary operation $\ell_1^\alpha=[\alpha,-]$). It turns out that this construction is a functor and, moreover, a comonad, and furthermore, coalgebras over this comonad are precisely operads $\mathcal{O}$ for which algebra structures can be twisted by solutions to the Maurer--Cartan equation. This construction has been used in a variety of contexts (from algebraic topology and deformation theory to, more recently, stochastic PDEs).