  Given a complex space $X$, we cosidered the problem of finding a {\it
hyperbolic model} of $X$. This is an object $\ip(X)$ with a morphism $i:X\to
\ip(X)$ in such a way that $\ip(X)$ is ``hyperbolic'' in a suitable sense and
$i$ is as close as possible to be an isomorphism. Using the theory of model
categories, we found a definition of hyperbolic simplicial sheaf (for the
strong topology) that extends the classical one of Brody for complex spaces. We
prove the existence of hyperbolic models for any simplicial sheaf. Furthermore,
the morphism $i$ can be taken to be a cofibration and an affine weak
equivalence (in an algebraic setting, Morel and Voevodsky called it an $\aff$
weak equivalence). Imitating one possible definition of homotopy groups for a
topological space, we defined the {\it holotopy} groups for a simplicial sheaf
and showed that their vanishing in ``positive'' degrees is a necessary
condition for a sheaf to be hyperbolic. We deduce that if $X$ is a complex
space with a non zero holotopy group in positive degree, then its hyperbolic
model (that in general will only be a simplicial sheaf) cannot be weakly
equivalent to a hyperbolic complex space (in particular is not itself
hyperbolic). We finish the manuscript by applying these results and a {\it
topological realization functor}, constructed in the previous section, to prove
that the hyperbolic models of the complex projective spaces cannot be weakly
equivalent to hyperbolic complex spaces.
