I have a feeling that this may be a very easy question for some people on MO, but it isn't for me.

Take a finite pointed set $X$, with $*$ the base-point. Build a cosimplicial set which in degree $n \ge 1$ is $X^n$ (the cartesian product of $n$ copies of $X$), and in degree $0$ is $\{ * \}$; the cofaces are:

$d^0(x_1, \ldots, x_n) = (*, x_1, \ldots, x_n)$

$d^i(x_1, \ldots, x_n) = (x_1, \ldots, x_i, x_i, x_{i+1}, \ldots, x_n)$

$d^{n+1}(x_1, \ldots, x_n) = (x_1, \ldots, x_n, *)$

Now apply the functor "free $k$-module on", where $k$ is your favorite ring. You get a cosimplicial $k$-module $A^*$, so you can build the associated cochain complex where the differential is the alternating sum of the cofaces. Note that $A^n = (A^1)^{\otimes n}$.

What is the cohomology of this complex?

Ideally someone will say something like "this is the cobar construction, it computes the cohomology of the loop space on the discrete space $X$, so the cohomology is $0$ in positive degrees", or something close. And it would be awesome. (The buzzword "cobar construction" seems to show up a lot among the papers I've skimmed through online.)

Thank you so much for your help!

Pierre