Categorified Fredholm Modules and Chern Characters

In this paper, we continue our program of systematic categorification of the Noncommutative Differential Geometry of Connes. We replace a ring with a small $\mathbb C$-linear category, seen as a ring with several objects in the sense of Mitchell. We introduced Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $\mathcal C$. We show that this categorified Chern character depends only on the homotopy class of the Fredholm module and is well-behaved with respect to the periodicity operator in cyclic cohomology.