Model structures and the Oka Principle

By F. Lárusson. 2003

71 citations recorded by the source corpus.

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 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8bcb9f68db82351af88bdee0591879eae36b18c

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Citations in the corpus

The spectrum of prime ideals in tensor triangulated categories (2004, background): “Always hidden in the shadow of every newly born Quillen model structure, which also entered complex geometry (see Lárusson [10]), triangulated categories notoriously appeared at the front line of the motivic battle, where Voevodsky won his titles of glory.”
Survey of Oka theory (2010, background): “The main results of [47] along with Theorem 6 of [48] can be summarized as follows.”
Holomorphic flexibility properties of complex manifolds (2004, unknown): “[33], [34], [35], [8], [36], [20], [53], [40], [42], [73], [43], [28], [29], [30], [22], [51], [52]).”