  Let $\Sigma$ be a finite regular cell complex with $\emptyset \in \Sigma$,
and regard it as a partially ordered set (poset) by inclusion. Let $R$ be the
incidence algebra of the poset $\Sigma$ over a field $k$. Corresponding to the
Verdier duality for constructible sheaves on $\Sigma$, we have a dualizing
complex $w \in D^b(mod_{R \otimes_k R})$ giving a duality functor from
$D^b(mod_R)$ to itself. $w$ satisfies the Auslander condition. Our duality is
somewhat analogous to the Serre duality for projective schemes ($\emptyset$
plays a similar role to that of "irrelevant ideals"). If $H^i(w) \ne 0$ for
exactly one $i$, then the underlying topological space of $\Sigma$ is
Cohen-Macaulay (in the sense of the Stanley-Reisner ring theory). The converse
also holds when $\Sigma$ is a simplicial complex.
  $R$ is always a Koszul ring with $R^! \cong R^op$. The relation between the
Koszul duality for $R$ and the Verdier duality is discussed. This result is a
variant of a theorem of Vybornov. The Mobius function of the poset
$\hat{\Sigma}$ is also discussed.
