  Let $\mathcal{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus
$g$ covers of $\mathbb{P}^1$. We define the tautological ring of
$\mathcal{H}_{k,g}$ and we show that all Chow classes, except possibly those
supported on the locus of "factoring covers," are tautological up to
codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also
used in our subsequent work, wherein we prove new results about the structure
of the Chow ring for $k \leq 5$.
