Given a ring $R$, we have an adjunction of the forgetful functor $CAlg_R\to Set$ and the free algebra functor $Set\to CAlg_R$, giving us an endofunctor $$T:CAlg_R\to CAlg_R, S\mapsto R[S].$$
In general any adjuction of a forgetful functor $\mathcal{C} \to Set$ and a free functor $Set\to \mathcal{C}$ yields such an endofunctor $\mathcal{C}\to \mathcal{C}$.
This in particualr yields $$\ldots R[R[S]]\rightrightarrows R[S]\to S$$ which we call a free resolution. What can we say about $\text{colim} T^i(S)$? When is this isomorphic to $S$? What can we say for a general bar-resolution coming from a monad?