Example 14.34.4. Let R be a ring. As an example of the above we can take i : \text{Mod}_ R \to \textit{Sets} to be the forgetful functor and F : \textit{Sets} \to \text{Mod}_ R to be the functor that associates to a set E the free R-module R[E] on E. For an R-module M the simplicial R-module X(M) will have the following shape
X(M) = \left( \ldots \xymatrix{ R[R[R[M]]] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & R[R[M]] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & R[M] \ar@<0ex>[l] } \right)
which comes with an augmentation towards M. We will also show this augmentation is a homotopy equivalence of sets. By Lemmas 14.30.8, 14.31.9, and 14.31.8 this is equivalent to asking M to be the only nonzero cohomology group of the chain complex associated to the simplicial module X(M).
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