  We categorify the theory developed by Moy-Prasad in [MP94]. More precisely,
we define a depth filtration on any category with an action of the loop group
$G((t))$ and prove a $2$-categorical generation statement inspired by the
theory of unrefined minimal K-types. Using our generation theorem, we compute
the depth filtration on the category of Whittaker sheaves and on the category
of Kac-Moody modules. On the Whittaker side, we show that it recovers and
extends the filtration constructed by Raskin in [Ras16]. For KM modules, our
computation encodes several new localization statements, as well as confirming
a conjecture of Chen-Kamgarpour.
