  Motivated by a recently established result saying that within the class of
bivariate Archimedean copulas standard pointwise convergence implies weak
convergence of almost all conditional distributions this contribution studies
the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with
$d \geq 3$ and proves the afore-mentioned implication with respect to
conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to
pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as
by-product of working with conditional distributions (Markov kernels) -
alternative simple proofs for the well-known formulas for the level set masses
$\mu_C(L_t)$ and the Kendall distribution function $F_K^d$ as well as a novel
geometrical interpretation of the latter are provided. Viewing normalized
generators $\psi$ of $d$-dimensional Archimedean copulas from the perspective
of their so-called Williamson measures $\gamma$ on $(0,\infty)$ is then shown
to allow not only to derive surprisingly simple expressions for $\mu_C(L_t)$
and $F_K^d$ in terms of $\gamma$ and to characterize pointwise convergence in
$\mathcal{C}_{ar}^d$ by weak convergence of the Williamson measures but also to
prove that regularity/singularity properties of $\gamma$ directly carry over to
the corresponding copula $C_\gamma \in \mathcal{C}_{ar}^d$. These results are
finally used to prove the fact that the family of all absolutely continuous and
the family of all singular $d$-dimensional copulas is dense in
$\mathcal{C}_{ar}^d$ and to underline that despite of their simple algebraic
structure Archimedean copulas may exhibit surprisingly singular behavior in the
sense of irregularity of their conditional distribution functions.
