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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$\begingroup$ Do you mean singular simplicial set instead of nerve? The nerve is usually a functor from categories to simplicial sets, not from spaces. $\endgroup$– Fernando MuroCommented Dec 22, 2011 at 0:02
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$\begingroup$ Nerve in a general sense: ncatlab.org/nlab/show/nerve+and+realization $\endgroup$– Jiarui FeiCommented Dec 22, 2011 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.
answered Dec 22, 2011 at 0:04
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2$\begingroup$ ... 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! $\endgroup$ Commented Dec 22, 2011 at 3:11