I'm reading a Bousfield and Kan's book and I'm curious could homotopical approach to homology be extended to cohomology?
I will elaborate a bit: homology with coefficients in commutative ring $R$ could be defined using homotopy groups of certain simplicial set ($R$-module to be precise): $$\widetilde{H}_*(X;R)=\pi_*(R \otimes X/R\otimes pt)$$ where $R$ is regarded as a simplicial module and $X$ is a simplicial set by default.
So my question here is could we define cohomology in a similar way? And if we can (which I am almost sure of) how this definition would interact with the machinery of completions and localizations developed by Bousfield and Kan?
My guess is that we should take something like $\widetilde{H}^*(X)=\pi_*($Hom$(X;R)/$Hom$(pt;R))$ under previous assumptions.
UPD: following remarks by Connor Malin I should clarify that $X$ here is a pointed simplicial set.
$R\otimes X$ is a "free simplicial $R$-module" generated by $X$ (i.e. $(R\otimes X)_n$ is a free $R$-module generated by $n$-simplices of $X$ with face and degeneracy operators induced from X).
I assume since you are using reduced cohomology that your $X$ is pointed. In this case it is true that $\bar{H}^k(X)=\pi_0(Hom(X,S^k R))$, where $S^k$ denotes shifting the chain complex up $k$ indices, because cohomology is represented by $K(R,*)$ and $S^k R$ by Dold-Kan has its homotopy group $R$ in dimension $k$ and $0$ elsewhere. Taking higher homotopy groups of this space just gives us lower cohomology groups.