Homotopy theory and complex geometry

homotopy theory provides a framework in which any site S can be embedded and its objects studied. In this context, objects of the site appear distinguished among more general ones. The purpose of this framework is to create an environment where the site S and supplementary “combinatorial/homotopical” data can blend together. Such a blending is achieved by a well-established mathematical procedure, the localization. The challenge in using this framework in practice is to find close ties between these general objects and the ones in S. I will describe the results that G. Tomassini and I have obtained on these topics.