Consider $p:E\rightarrow B$ to be a fibration of simplicial sets. I want to see that if $p$ is an epimorphism and $E$ is fibrant then $B$ is also fibrant.
I have come to the conclusion that to solve this problem I just need to find a lift
$\require{AMScd}$ \begin{CD} . @>{}>> E\\ @VVV @VVV\\ \Lambda^k_n @>{}>>B \end{CD}
from $\Lambda^k_n$ to $E$. Now by assumption we know that $p:E\rightarrow B$ is a fibration. Now I belive that $\cdot \rightarrow \Lambda^k_n$ is a trivial cofibration , since it's an homotopy equivalence on the geometric realizations and a monomorphism. Now due to well know facts on the model structure we have that such a lift exists. I wanted to make sure that all the facts that I am using , namely that $\cdot \rightarrow \Lambda^k_n$ is a trivial cofibration are correct. Also that if we have an epimorphism $p:E\rightarrow B$ then the maps $p_n:E_n\rightarrow B_n$ are surjective.
Any insight is appreciated, thanks in advance.