  In this note we present a work in progress whose 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 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 X, which is a derived scheme
whose theory of functions is closely related to cyclic homology of X. 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.
