Extended Hyperbolicity
  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. 
Simone Borghesi and Giuseppe Tomassini