Say in the most classical case, we probe a topological space $X$ by the n-simplices $\Delta^n$ by using the nerve functor $Hom_{Top}(-,X)$. Is another functor $Hom_{Top}(X,-)$ of any use, or is there a dual notion for the nerve functor? Why its left adjoint geometric realization has a dual called totalization? Thank you so much?