# Do comonads always induce cosimplicial objects? Vice-Versa?

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?

Thanks! Jon

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One note, I demand that $X$ be $\bot$-projective, but that is not necessary. If $X$ is $\bot$-projective then we have an "aspherical" simplicial object. – Jon Beardsley Oct 12 '11 at 15:42
What one specifically gets from a comonad $L$ on $\mathcal{C}$ is a functor $F:\mathcal{C}\to \mathcal{C}^{\boldsymbol{\Delta}^{op}}$. So I doubt one could generally take a single simplicial object and get a comonad back out of it since, as you suppose, that's a lot of information to ask from one simplicial object. I have no idea about obtaining a comonad from, say, some sufficiently nice subcategory of $\mathcal{C}^{\boldsymbol{\Delta}^{op}}$, but I hope someone has an answer to that! – Malice Vidrine Jun 26 at 7:14