Indirect method:

Monoids are categories with one object. A simple calculation shows that the inclusion of categories **N** into **Z** has contractible homotopy fiber (which is the nerve of the category whose objects are the arrows of the one object category **Z** and whose arrows are commutative triangles of **Z** mediated by the elements of **N**). Thus Quillen's theorem A yields a homotopy equivalence of the corresponding nerves arising from the inclusion of underlying categories.

Direct method:

Consult this paper by Ken Brown: "[The Geometry of Rewriting Systems][1]"

You need only the simplest version of his method. With it one can show that the nerve of **N** and the nerve of **Z** have cellular models which differ only by collapses of simplices and thus have the same (simple) homotopy type.


  [1]: http://www.math.cornell.edu/~kbrown/scan/1992.0000.0137.pdf