Examples
The primordial example is the classifying space of a discrete group G. We regard G as a category with one object whose endomorphisms are the elements of G. Then the k-simplices of N(G) are just k-tuples of elements of G. The face maps act by multiplication, and the degeneracy maps act by insertion of the identity element. If G is the group with two elements, then there is exactly one nondegenerate k-simplex for each nonnegative integer k, corresponding to the unique k-tuple of elements of G containing no identities. After passing to the geometric realization, it is not hard to see that this k-tuple can be identified with the unique k-cell in the usual CW structure on infinite-dimensional real projective space. The latter is the most popular model for the classifying space of the group with two elements. See (Segal 1968) for further details and the relationship of the above to Milnor's join construction of BG.
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