  By careful analysis of the comparison map from a simplicial set to its image
under Kan's ex-infinity functor we obtain a new and combinatorial proof that it
is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a
strong anodyne extension. From this description we are able to quickly deduce
some basic facts about ex-infinity and hence provide a new construction of the
Kan-Quillen model structure on simplicial sets, which avoids the use of
topological spaces or minimal fibrations.
