  We study reductions of 6d theories on a $d$-dimensional manifold $M_d$,
focusing on the interplay between symmetries, anomalies, and dynamics of the
resulting $(6-d)$-dimensional theory $T[M_d]$. We refine and generalize the
notion of "polarization" to "polarization on $M_d$," which serves to fix the
spectrum of local and extended operators in $T[M_d]$. Another important feature
of theories $T[M_d]$ is that they often possess higher-group symmetries, such
as 2-group and 3-group symmetries. We study the origin of such symmetries as
well as physical implications including symmetry breaking and symmetry
enhancement in the renormalization group flow. To better probe the IR physics,
we also investigate the 't Hooft anomaly of 5d Chern-Simons matter theories.
The present paper focuses on developing the general framework as well as the
special case of $d=0$ and 1, while an upcoming paper will discuss the case of
$d=2$, $3$ and $4$.
