Abstract
We replace a ring with a small ℂ-linear category 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 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 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.
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