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In this note we present a work in progress whose main purpose is to establisha categorified version of sheaf theory. We present a notion of derivedcategorical sheaves, which is a categorified version of the notion of complexesof sheaves of modules on schemes, as well as its quasi-coherent and perfectversions. We also explain how ideas from derived algebraic geometry and highercategory theory can be used in order to construct a Chern character for thesecategorical sheaves, which is a categorified version of the Chern character forperfect complexes with values in cyclic homology. Our construction uses in anessential way the derived loop space of a scheme X, which is a derived schemewhose theory of functions is closely related to cyclic homology of X. This workcan be seen as an attempt to define algebraic analogs of elliptic objects andcharacteristic classes for them. The present text is an overview of a work inprogress and details will appear elsewhere.

A note on Chern character, loop spaces and derived algebraic geometry