Extended hyperbolicity (Q5894311)

The aim of this paper is to generalize the useful notion of Brody hyperbolicity already known for complex spaces. (A complex space \(X\) is Brody hyperbolic if there is no non-constant holomorphic map \(\mathbb{C}\to X\).) So the authors extend the definition to the case of simplicial sheaves of sets over the site of complex spaces with the strong topology. The topological background can be found in [\textit{F. Morel} and \textit{V. Voevodsky}, Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)]. Imitating one possible definition of homotopy groups for a topological space, they define the homotopy groups for a simplicial sheaf and show that their vanishing in ``positive'' degrees is a necessary condition for a sheaf to be Brody hyperbolic.