  We give necessary and sufficient conditions for an operator $A:X\to Y$ on a
Banach space having a shrinking FDD to factor through a Banach space $Z$ such
that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove
that for every ordinal $\xi\in (0, \omega_1)\setminus\{\omega^\eta:
\eta<\omega_1\text{\ a limit ordinal}\}$, there exists a Banach space
$\mathfrak{G}_\xi$ having a shrinking basis and Szlenk index $\omega^\xi$ such
that for any separable Banach space $X$ and any operator $A:X\to Y$ having
Szlenk index less than $\omega^\xi$, $A$ factors through a subspace and through
a quotient of $\mathfrak{G}_\xi$, and if $X$ has a shrinking FDD, $A$ factors
through $\mathfrak{G}_\xi$.
