  Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith
cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex
for the Andr\'{e}-Quillen cohomology groups in which the $k$-invariants for
$\mathcal{X}$ take value. Given a map of simplicial categories
$\phi:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of
$\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to
study the obstruction to lifting $\phi$ to $P^{(n)}\mathcal{X}$. In particular,
an explicit description of this obstruction for the boundary of a cube can be
used to recover various higher homotopy invariants of $\mathcal{X}$.
