  We replace a ring with a small $\mathbb C$-linear category $\mathcal{C}$,
seen as a ring with several objects in the sense of Mitchell. We introduce
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 is homotopy invariant and is well-behaved with respect to the
periodicity operator in cyclic cohomology. For this, we also obtain a
description of cocycles and coboundaries in the cyclic cohomology of $\mathcal
C$ (and more generally, in the Hopf-cyclic cohomology of a Hopf module
category) by means of DG-semicategories equipped with a trace on endomorphism
spaces.
