  We consider feed-forward networks, that is, networks where cells can be
divided into layers, such that every edge targeting a layer, excluding the
first one, starts in the prior layer. A feed-forward system is a dynamical
system that respects the structure of a feed-forward network. The synchrony
subspaces for a network, are the subspaces defined by equalities of some cells
coordinates, that are flow-invariant by all the network systems. The
restriction of each network system to each synchrony subspace is a system
associated with a smaller network, which may be, or not, a feed-forward
network. The original network is then said to be a lift of the smaller network.
We show that a feed-forward lift of a feed-forward network is given by the
composition of two types of lifts: lifts that create new layers and lifts
inside a layer. Furthermore, we address the lifting bifurcation problem on
feed-forward systems. More precisely, the comparison of the possible
codimension-one local steady-state bifurcations of a feed-forward system and
those of the corresponding lifts is considered. We show that for most of the
feed-forward lifts, the increase of the center subspace is a sufficient
condition for the existence of additional bifurcating branches of solutions,
which are not lifted from the restricted system. However, when the bifurcation
condition is associated with the internal dynamics and the lifts occurs inside
an intermediate layer, we prove that the existence of bifurcating branches of
solutions that are not lifted from the restricted system does depend
generically on the particular feed-forward system.
