  We discuss the most general field equations for cosmological spacetimes for
theories of gravity based on non-linear extensions of the non-metricity scalar
and the torsion scalar. Our approach is based on a systematic
symmetry-reduction of the metric-affine geometry which underlies these
theories. While for the simplest conceivable case the connection disappears
from the field equations and one obtains the Friedmann equations of General
Relativity, we show that in $f(\mathbb{Q})$ cosmology the connection
generically modifies the metric field equations and that some of the connection
components become dynamical. We show that $f(\mathbb{Q})$ cosmology contains
the exact General Relativity solutions and also exact solutions which go
beyond. In $f(\mathbb{T})$~cosmology, however, the connection is completely
fixed and not dynamical.
