Chern character, loop Spaces and derived algebraic geometry

Contributo su invito. Abstract- In this note we present a work in progresswhose main purpose is to establish a categorified version of sheaf theory.We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme , which is a derived scheme whose theory of functions is closely related to cyclic homology of . This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere (see TV1 and TV2).