Take the 2-minute tour ×  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?  share|improve this
question 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  1 Answer 1  up vote 3 down
vote accepted  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.  share|improve this answer ...
or if your willing to go further and drink the derived cool-aid,
you can just use the s