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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?

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Do you mean singular simplicial set instead of nerve? The nerve is usually a functor from categories to simplicial sets, not from spaces. –  Fernando Muro Dec 22 '11 at 0:02
Nerve in a general sense: ncatlab.org/nlab/show/nerve+and+realization –  Jiarui Fei Dec 22 '11 at 16:53

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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.

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... or if your willing to go further and drink the derived cool-aid, you can just use the spectrum Hom(X,K) which is a module for the ring spectrum K=HZ and represents the total singular cohomology complex. Or even better, let K range over all spectra. Then you get all generalized cohomology theories this way! –  Chris Schommer-Pries Dec 22 '11 at 3:11

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