The book says:
Definition 1.4.1. A complex $C$ is a called split if there are maps $s_n : C_{n+1} \to C_{n+1}$ such that $d = dsd$. The maps $s_n$ are called splitting maps. If in addition $C$ is acyclic (exact as a sequence), we say that $C$ is split exact.
My question is what is the precise formula for $d = dsd$ in terms of indexes? Is it $d_{n+1}s_n d_{n+2} = d_{n+1}$ ? It seems weird. Why not have $s_n : \ C_n \to C_n$, then it would be $d_n s_n d_{n+1} = d_n$ which is cleaner.\
Cancel, that the maps as I've stated them don't make sense (are not defined). Please illuminate my error or the book's error. Thanks.
As @Hoot says, the splitting should be a map $s_n : C_n \to C_{n+1}$ (and it is indeed what's written in my version of the book). Note that in the formula you have written, the domain of the RHS is not equal to the domain of the LHS...
If you want to remember how the splitting map works, look at the definition of a chain homotopy below (in the book). A chain homotopy between the identity and the zero map of $C_*$ is a map $s$ such that $\operatorname{id}_C = ds + sd$ (and in this formula it should not be possible to confuse indices); if you apply $d$ either on the left or on the right, you get $d = dsd$ (because $d^2 = 0$), ie. $s$ is in particular a splitting.