Say in the most classical case, we probe a topological space $X$ by the nsimplices $\Delta^n$ 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?

If you're willing to work in an appropriate homotopy category instead of Top, you can take K to be the Eilenberg Maclane spectrum so that Hom(X,K) is (for reasonable X) the singular cohomology of X. 

