Vtyjkyuk

I am reading about the bar/cobar construction in the book Algebraic Operads. The differential on the bar construction of a augmented dga algebra $A$ is a sum of two differentials $d_1+d_2$ where $d_1$ is somehow induced from the differential $d_A$ on $A$. I know we get a differential on $A^otimes n$ by taking $sum_i 1^otimes i-1otimes d_Aotimes 1^otimes n-i$. But this is only a differential on $barA^otimes n$ if $d_A(barA)subset barA$ (where $barA$ is the kernel of the augmentation map). My question is if this is true in general and if not how do we get a differential on $barA^otimes n$?

Much grateful for any answer=)

An augmentation of a nonnegatively graded chain complex $A$ is defined as a chain map to the complex that is $k$ in degree zero and $0$ otherwise (see section 1.5.5 in Loday's Algebraic Operads). Thus, it commutes with the boundary map $d_A$ by definition. As mentioned in the comments, this means the augmentation ideal is preserved by $d_A$. So there's no problem in defining the differential in the bar construction as you have.