I tried to convince myself recently that the geometric realization of the nerve of the 1-simplex was homeomorphic to the 1-simplex, but I got a bit stuck. I now realize that I'm not actually sure whether it's even true—can anyone give a proof one way or the other?
To be totally clear, by "nerve" I mean specifically the presheaf on the simplex category given by probing a topological space with the standard simplices—not the nerve of a category or anything.