Abstract. 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 results to the effect 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. 1. Introduction. This paper, like its predecessor [L], is about model structures in complex analysis. Model structures are good for many things, but here we view them primarily as a tool for studying lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. More precisely, model structures provide a framework for investigating two classes of holomorphic maps such that the first has the right lifting property with respect to the second and the second has the left lifting property with respect to the first in the absence of topological obstructions. (It is more useful, actually, to consider homotopy lifting properties rather than plain lifting properties.) We seek to make the maps in the first class into fibrations and those in the second class into cofibrations, with weak equivalences being understood in the topological sense. The machinery of abstract homotopy theory can then be applied. We equip the category of Stein manifolds in a natural way with a simplicial structure and a compatible topology, turning it into a simplicial site, and embed the category of complex manifolds into the simplicial category of prestacks on this site. By a prestack we
Lárusson REF has recently used a flasque model structure in the context of complex analysis.
Model structures and the Oka principle