The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known
$\textsf{W}[2]$
-hardness results, we show that these problems belong to
$\textsf{A}[2]$
,
$\textsf{W}[\textsf{P}],$
and
$\textsf{WNL}$
. This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class
$\textsf{W}[\textsf{Sync}]$
as a proper home for these (and more) problems. We present quite a number of problems that belong to
$\textsf{W}[\textsf{Sync}]$
or are hard or complete for this new class.