Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$.
- Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of $|X|$).
- We can compose $X$ and functor of simplicial chains $C_* \circ X$, and we get a simplicial complex (for safety it is better consider rational coefficients). Via Dold-Kan correspondence we get a double complex $DK( C_* \circ X )$. Finally you consider total complex of this double complex $Tot \ DK( C_* \circ X )$ and count cohomology.
Question: Do these notions coincide?
I am not looking for pathological counterexamples. If it is needed to add some sensible assumptions, then do it.