Extended hyperbolicity

The concept of Brody hyperbolicity is interpreted in terms of homotopy theoretic structures. We extend the definition of Brody hyperbolicity to simplicial sheaves of sets over the site of complex spaces with the strong topology. 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 Brody hyperbolic. A partial converse to this theorem is proved at the end of the paper. We deduce that if $X$ is a complex space with a non zero holotopy group in positive degree, then $X$ cannot be weakly equivalent (in a particular sense) to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results along with a {\it topological realization functor}, constructed in the previous section, to prove that complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces.