Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology

Abstract

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.

This paper supersedes and replaces both arXiv:1912.12658 [math.CT] and also arXiv:1901.09580 [math.CT]