  Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it
$(\alpha,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting
a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing
the incident hinges and edges among the subvertices. A detachment is {\it fair}
if the degree of vertices and multiplicity of edges are shared as evenly as
possible among the subvertices within the whole hypergraph as well as within
each color class. In this paper we solve an open problem from 70s by finding
necessary and sufficient conditions under which a $k$-edge-colored hypergraph
$\mathcal G$ has a fair detachment in which each color class is connected.
Previously, this was not even known for the case when $\mathcal G$ is an
arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our
theorem by proving a variety of new results on hypergraph decompositions, and
completing partial regular combinatorial structures.
