I have seen deformation problems given by commutative squares
$
\begin{array}.X & \rightarrow & \tilde{X}\\
\downarrow & & \downarrow\\
\mathrm{Spec}\, A & \rightarrow & \mathrm{Spec}\, B\\
\end{array}
$
where $X$ is a scheme and $A$ and $B$ are local Artin $\mathbf{k}$-algebras.
My question is, how does one translate a geometric deformation problem, say deformations of a uniform lattice (discrete Lie group) $\Gamma < G$ acting properly discontinuously (cocompactly, whatever) on a homogeneous space $G/H$, into a deformation diagram/functor that resembles the above diagram?
