2 problems

Let $\mathcal{O}$ be an operad in $\mathcal{R}$-modules with $\mathcal{O}[0]$ the null object, and factor the canonical map $\mathcal{O}\to\tau_1\mathcal{O}$ as a cofibration follo…

Let $\mathcal{O}$ be an operad, and let $\mathsf{K}$ be the comonad associated to the adjunction comparing $\mathcal{O}$-algebras with $\mathsf{K}$-coalgebras. A homotopy pro-nilpo…