\chapter{\texorpdfstring{$A$}{A} and \texorpdfstring{$K$}{K} form a Quillen pair}
\chapter[A and K form a Quillen pair]{$A$ and $K$ form a Quillen pair}
\label{sec:a-k-quillen-pair}
We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration.