- Given a monad $M:C\to C$ we can construct a cobar resolution from it directly as a functor $\Delta\to [C,C]$
- Given a DG-coalgebra $(C,d)$ we can construct a cobar resolution $\Omega C$ of it as follows: $\Omega C = T(s^{-1}\bar C)$ here $s^{-1}$ is a desuspension and $\bar C$ is a coaugmentation ideal of $C$ with differential $d_{\Omega}(s^{-1} c)=-s^{-1}d(c)+(s^{-1}\otimes s^{-1})\bar\Delta(c)$, here $\bar\Delta$ is a reduced comultiplication.
I'm trying to understand how to get (2) from (1). We start with monad $\bar C\otimes (-)\circ U$ on $\bar C$-comodules ($U$ is a forgetful functor), take corresponding cobar construction, evaluate it at the ground ring and normalize it, we got a cochain complex with $\bar C^{\otimes n}$ in dimension $n$. We can sum it up and get a tensor algebra, desuspension comes from putting degree shifting under tensor products. Is this correct? Now I have troubles with understanding $d_{\Omega}$. I can vaguely understand that comultiplication part comes from cochain differential, but have no ideas where from $-s^{-1}d$ part comes from. Multiplication on $\Omega C$ comes from multiplication on tensor algebra, but is there a way to see it from cobar resolution directly? Appreciate any help.