Cycles over DGH-semicategories and pairings in categorical Hopf-cyclic cohomology

Abstract

Let H be a Hopf algebra and let D H be a Hopf-module category. We introduce the Hopf-cyclic cohomology groups HC • H (D H , M ) of a Hopf-module category D H with coefficients in a stable anti-Yetter Drinfeld (SAYD) module M over H. For an H-module coalgebra C acting on D H , we construct a pairing HC q H (C, M ) ⊗ HC p H (D H , M ) → HC p+q (D H ) with the Hopf-cyclic cohomology of C with coefficients in M . We describe the cocycles Z • H (D H , M ) and the coboundaries B • H (D H , M ) as characters of categorified cycles and vanishing cycles over D H . Using this formalism, we obtain a pairing HC p (C) ⊗ HC q (C ′ ) → HC p+q (C ⊗ C ′ ) for small k-linear categories C and C ′ .

Categorified Fredholm Modules and Chern Characters

“…We first recall from [1] the construction of universal DG-semicategories and the notion of closed graded traces on DG-semicategories. In essence, a semicategory is a "category without units."…”

“…for all f ∈ Hom n−1 S (X, X), g ∈ Hom i S (Y, X), g ′ ∈ Hom j S (X, Y ) and i + j = n. Proposition 2.5. (see [1,Proposition 5.13]) Let C be a small C-category and let T be a closed graded trace of dimension n on the universal DG-semicategory ΩC. Let CN • (C) = {Hom(CN n (C), C)} n≥0 be the dual of the cyclic nerve of C. Define φ ∈ CN n (C) by setting…”

“…We replace a ring with a small C-linear category, seen as a ring with several objects in the sense of Mitchell [13]. In [1], we have described the cocycles and coboundaries that determine the cyclic cohomology H • λ (C) by extending Connes' original construction of cyclic cohomology in terms of cycles and closed graded traces on differential graded algebras. The key objective of [1] is to describe cyclic cocycles and coboundaries in terms of closed graded traces over a differential graded semicategory.…”

“…In [1], we have described the cocycles and coboundaries that determine the cyclic cohomology H • λ (C) by extending Connes' original construction of cyclic cohomology in terms of cycles and closed graded traces on differential graded algebras. The key objective of [1] is to describe cyclic cocycles and coboundaries in terms of closed graded traces over a differential graded semicategory. In fact, we showed in [1] that each cyclic cocycle in Z n λ (C) is obtained as the character of a n-dimensional cycle over C, while every cyclic coboundary in B n λ (C) is given by the character of a "vanishing cycle."…”

“…The key objective of [1] is to describe cyclic cocycles and coboundaries in terms of closed graded traces over a differential graded semicategory. In fact, we showed in [1] that each cyclic cocycle in Z n λ (C) is obtained as the character of a n-dimensional cycle over C, while every cyclic coboundary in B n λ (C) is given by the character of a "vanishing cycle." This description of cyclic cohomology is best suited for the study of 'categorified Fredholm modules' that we now develop in this paper.…”