TITLE: Kan fibrations and surjectivity
QUESTION [7 upvotes]: I have a basic question on the usual model structure on simplicial sets.

What is the relation between being a Kan (trivial maybe ?) fibration and
  surjectivity ?

Surjectivity here means either surjectivity on the components, or surjectivity at each level of the simplicial set, or other interesting notions. 
In Simplicial homotopy theory of Goerss and Jardine, they see at a moment, "since trivial fibrations are surjective, the result follows" (Proposition 3.3 of Chapter II). Is this surjectivity on the components ? 
Also, if you have a reference to point me too that would be great too, I haven't found much neither in Simplicial homotopy theory nor in others similar books.

REPLY [2 votes]: Let me just record here for clarity the actual conclusion from Aaron's answer.

If $f: X \to Y$ is a Kan fibration and $\sigma \in Y_n$ is a simplex in $Y$, then $\sigma$ is in the image of $f$ if and only if the path component of $\sigma$ is in the image of $f$.

Of course, there are plenty of maps satisfying the surjectivity condition which are not Kan fibrations.