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

TL;DR

This paper generalizes the concept of Fredholm modules and the Chern character to small categories, establishing homotopy invariance and compatibility with cyclic cohomology periodicity, enriching the categorical framework of noncommutative geometry.

Contribution

It introduces Fredholm modules over categories and constructs a categorified Chern character with invariance properties, extending classical concepts to categorical settings.

Findings

Defined Fredholm modules over categories.

Constructed a homotopy-invariant Chern character.

Described cyclic cohomology cocycles via DG-semicategories.

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.