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

Balodi, Mamta; Banerjee, Abhishek

Comptes Rendus. Mathématique, Tome 361 (2023) no. G3, pp. 617-652

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.

Classification : 18E05, 47A53, 53C99, 58B34

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