  We introduce the categories of infinitesimal Hopf modules and bimodules over
an infinitesimal bialgebra. We show that they correspond to modules and
bimodules over the infinitesimal version of the double. We show that there is a
natural, but non-obvious way to construct a pre-Lie algebra from an arbitrary
infinitesimal bialgebra and a dendriform algebra from a quasitriangular
infinitesimal bialgebra. As consequences, we obtain a pre-Lie structure on the
space of paths on an arbitrary quiver, and a striking dendriform structure on
the space of endomorphisms of an arbitrary infinitesimal bialgebra, which
combines the convolution and composition products. We extend the previous
constructions to the categories of Hopf, pre-Lie and dendriform bimodules. We
construct a brace algebra structure from an arbitrary infinitesimal bialgebra;
this refines the pre-Lie algebra construction. In two appendices, we show that
infinitesimal bialgebras are comonoid objects in a certain monoidal category
and discuss a related construction for counital infinitesimal bialgebras.
