  In this paper, we give the categorification of Leibniz algebras, which is
equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure
of omni-Lie 2-algebras introduced in \cite{omniLie2} as well as twisted Courant
algebroids by closed 4-forms introduced in \cite{4form}.
  We also prove that Dirac structures of twisted Courant algebroids give rise
to 2-term $L_\infty$-algebras and geometric structures behind them are exactly
$H$-twisted Lie algebroids introduced in \cite{Grutzmann}.
