  Given a functor $p:E \rightarrow B$ and an object $e \in E$ , we define a
\emph{displacement} of $e$ along a morphism $\varepsilon: p(e) \rightarrow b$,
as a map $e \rightarrow \nabla_\varepsilon(e)$ satisfying a universal property
analogue to that of a \emph{cocartesian lifting} (pushforward) \emph{\`a la}
B\'enabou-Grothendieck-Street. There are many difficulties in geometry that
come from the fact that forgetful functors such as $p: Var(\mathbf{C})
\rightarrow Top $ don't have displacements of objects along arbitrary maps. And
this can be already seen abstractly, since the existence of a left adjoint to
$p$, can be reduced to the existence of all displacements of the initial
object. However some \emph{schematization functors} exist as approximations. In
a broader context, if $B$ is a model category and $p$ is a right adjoint, then
the right-induced model category on $E$ exists if and only if all displacements
along any trivial cofibration $\varepsilon$, are weak $p$-equivalences. In
these notes we provide some categorical lemmas that will be necessary for
future applications. The idea is to have a \emph{homotopy descent process} for
\emph{elementary displacements} when $p$ has a \emph{presentation} as a
$2$-pullback of a family $\{p_i: E_i \rightarrow B\}_{i\in J}$. When suitably
applied it should lead to techniques similar to Mumford's GIT through homotopy
theory (simplicial presheaves).
