Non-Kan Fibrations In the definition of a Kan fibration (on nlab), i.e. for a map $\pi:Y\to X$ of simplicial sets the inclusion of any horn into $Y$ always lifts to an inclusion of the filled in horn if that filled in horn includes into $X$, it seems hard to imagine, at least geometrically, a counterexample.  Is there some intuitive, visual way to see under what conditions this might not happen? Also, if my definition is not correct, please edit!
Thanks!
 A: (I'm sure by now you know of many examples, but in case someone else wanted an answer to this question...)
For one, any non-Kan complex has the property that $K \rightarrow \ast$ is not a Kan fibration. For example, any horn $\Lambda^n_i$ fails to be a Kan complex (for obvious reasons), and the nerve of any category that's not a groupoid will fail to be a Kan complex (this includes, as a special case, $\Delta^n = N([n])$).
Non-point examples include: Most functors $\mathcal{C} \rightarrow \mathcal{D}$ between categories do not induce Kan fibrations on nerves $N\mathcal{C} \rightarrow N\mathcal{D}$ (this happens if and only if you have a category both fibered and cofibered in groupoids!) [Also, to your point about different model structures, these are examples of "inner fibrations" that are not Kan fibrations... if you have a category cofibered in groupoids, you get "left fibrations" that are not Kan fibrations, etc.) 
Basically most maps arising in higher category theory are not Kan fibrations... This is usually a special thing and when it happens you're happy.
