  We discuss the so-called "simplifying assumption" of conditional copulas in a
general framework. We introduce several tests of the latter assumption for non-
and semiparametric copula models. Some related test procedures based on
conditioning subsets instead of point-wise events are proposed. The limiting
distribution of such test statistics under the null are approximated by several
bootstrap schemes, most of them being new. We prove the validity of a
particular semiparametric bootstrap scheme. Some simulations illustrate the
relevance of our results.
