  The purpose of this work is to provide details about the construction of the
Chern character for categorical sheaves mentioned in our previous work "Chern
character, loop spaces and derived algebraic geometry". For this, we introduce
and study the notion of rigid symmetric monoidal \infty-category. We show how
trace maps can be constructed in this higher categorical setting, and using a
recent work of Hopkins-Lurie we prove the existence of a "cyclic trace", which
is the main ingredient in the construction of the Chern character. Our Chern
character is then constructed for any pair (T,A), consisting of a \infty-topos
T and a stack of rigid symmetric monoidal \infty-categories A on T. We propose
two main applications of this construction. First of all, we show how to
recover the Chern character of perfect complexes on schemes, with values in
cyclic homology, and show how it can be extended to an interesting new Chern
character for Artin stacks. Our second application provides invariants of
famillies of dg-categories. A consequence of the existence of these invariants
is the construction of a Gauss-Manin connexion on the cyclic homology complex
of such a familly, generalizing previous constructions by Getzler and of
Dolgushev-Tamarkin-Tsygan. Another consequence is the construction of the
"character sheaf" associated to a representation of an algebraic group into a
dg-category, which is a categorification of the character function of a linear
representation. Finally, for a familly of saturated dg-categories we construct
a "secondary Chern character", taking values in a new cohomology theory called
"secondary cyclic homology".
