Verdier duality ($D_X$)

On bounded constructible complexes over locally compact stratified spaces, Verdier duality is $D_{X}E=R\mathcal{H}om(E,{\omega }_X^{\bullet })$, where ${\omega }_X^{\bullet }$ is the dualizing complex. For an oriented real $N$-manifold, $D_X{\mathbb{Q}}_X\cong {\mathbb{Q}}_X[N]$. Proper direct image commutes with this duality, and $i_x^{!}{D}_{X}E\cong D_{\{x\}}{i}_x^{\ast }E$. These properties exchange stalk support conditions with costalk conditions.