  There are infinitely many variants of the notion of Kan fibration that,
together with suitable choices of cofibrations and the usual notion of weak
equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model
category. The combinatorics underlying these fibrations is purely finitary and
seems interesting both for its own sake and for its interaction with homotopy
types. To show that these notions of fibration are indeed distinct, one needs
to understand how iterates of Kan's Ex functor act on graphs and on nerves of
small categories.
