So Charles Weibel, in his book "Introduction to Homological Algebra" discusses the idea of a cotriple or a comonad. I believe he say that, given a comonad $\bot$ and an object $X$ which is "$\bot$-projective," we can look at a cosimplicial object $\bot_\ast X$. I maybe have some of my co's mixed up here. In general, is there a way to get back from a (co)simplicial object to a (co)monad? It seems that there may just not be enough information there, making this uninteresting.
However, if this is the case, is there a way to check if a (co)simplicial object CAN be induced by a (co)monad?