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

Caract\`eres de Chern, traces \'equivariantes et g\'eom\'etrie alg\'ebrique d\'eriv\'ee