  We prove estimates for the level of distribution of the M\"obius function,
von Mangoldt function, and divisor functions in squarefree progressions in the
ring of polynomials over a finite field. Each level of distribution converges
to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root
cancellation as $q$ goes to $\infty$. These results in fact apply to a more
general class of functions, the factorization functions, that includes these
three. The divisor estimates have applications to the moments of $L$-functions,
and the von Mangoldt estimate to one-level densities.
