TITLE: Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
QUESTION [6 upvotes]: Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by    $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample?

The only things I can say are that it preserves limits because it is a right adjoint and that it preserves filtered colimits since each $\Delta^m$ is a finite CW-complex.

Does the functor $S$ preserve homotopy colimits? If not,    what is a counterexample?

Again I can only say that it preserves homotopy limits as because it is a right Quillen adjoint. This Quillen adjunction with left adjoint the realization functor however is also a Quillen equivalence. I wonder if it follows already from this fact, that the right adjoint preserves homotopy colimits (and that the left adjoint preserves homotopy limits).

REPLY [2 votes]: The first question: This is usually not the case. For example, you can find a counterexample from that the Yoneda functor does not preserve colimits. 
Since the colimit in $sSet$ is computed pointwise, one looks for $X_1, X_2, X_0, X_3=X_1\cup_{X_0} X_2$ such that $\hom(\Delta^1, X_i)$ is not a colimit diagram. Let 
$$X_1=[0,1],\quad X_2=[-1,0],\quad X_0=0,\quad X_3=[-1, 1].$$
Then it is clear that $X_3$ has more paths than paths of $X_1$ plus paths of $X_2$.