  Inspired by recent developments of quantum speed limit we introduce a
categorical energy of sheaves in the derived category over a manifold relative
to a microlocal projector. We utilize the tool of algebraic microlocal analysis
to show that with regard to the microsupports of sheaves, our categorical
energy gives a lower bound of the Hofer displacement energy. We also prove that
on the other hand our categorical energy obeys a relative energy-capacity type
inequality. As a by-product this provides a sheaf-theoretic proof of the
positivity of the Hofer displacement energy for disjointing the zero section
$L$ from an open subset $O$ in $T^*L$ , given that $L \cap O \neq \emptyset$.
