  We embed the category of complex manifolds into the simplicial category of
prestacks on the simplicial site of Stein manifolds, a prestack being a
contravariant simplicial functor from the site to the category of simplicial
sets. The category of prestacks carries model structures, one of them defined
for the first time here, which allow us to develop "holomorphic homotopy
theory". More specifically, we use homotopical algebra to study lifting and
extension properties of holomorphic maps, such as those given by the Oka
Principle. We prove that holomorphic maps satisfy certain versions of the Oka
Principle if and only if they are fibrations in suitable model structures. We
are naturally led to a simplicial, rather than a topological, approach, which
is a novelty in analysis.
