  For a fixed algebraic variety $X$, curve class $\alpha \in N_1(X)$, and genus
$g \in \mathbb N$, we consider the sequence of $S_n$ representations obtained
from the homology of the Kontsevich space of stable maps to $X$, $\bar
M_{g,n}(X,\alpha)$. Using the category of finite sets and surjections, we prove
a representation stability theorem that governs the behavior of this sequence
of representations for $n$ sufficiently large.
