[Paper Review] Some invariance properties of cyclic cohomology with coefficients

Ilya Shapiro|arXiv (Cornell University)|Nov 4, 2016

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper introduces a derived definition of cyclic cohomology with coefficients in monoidal categories, enhancing invariance properties such as Morita and 2-Morita invariance. The key contribution is Theorem 4.31, which establishes a general Shapiro-type lemma for cyclic cohomology across different monoidal categories via a 2-functorial framework, resolving limitations of prior definitions in non-semisimple settings.

ABSTRACT

In this paper, we further explore the conceptual approach to cyclic cohomology with coefficients. In particular we give a derived version of the definition with better invariance properties. We show that the new definition agrees with the old under certain conditions and we prove, for the new definition, both its Morita invariance and its $2$-Morita invariance, under suitable interpretations of these terms. More generally, we prove a version of Shapiro's lemma for cyclic cohomology with coefficients.

Motivation & Objective

• To develop a derived, homologically robust definition of cyclic cohomology with coefficients in monoidal categories that improves invariance properties.
• To establish Morita invariance and 2-Morita invariance for the new definition under suitable categorical interpretations.
• To generalize Shapiro’s lemma to cyclic cohomology with coefficients in the context of 2-functors between module 2-categories.
• To clarify the role of contramodules and symmetric 2-contratraces in the framework, particularly in non-semisimple Hopf algebra settings.
• To unify and extend previous approaches to Hopf-cyclic cohomology, including type A and B theories, via categorical and higher-categorical structures.

Proposed method

• The paper constructs precocyclic objects in abelian monoidal categories using a new approach distinct from [HKS], enabling derived homological properties.
• It introduces a derived version of cyclic cohomology that preserves invariance under 2-Morita equivalence, replacing cocyclic objects with precocyclic ones.
• The method relies on the categorical notion of a 2-contratrace, with automorphisms induced via Yoneda arguments on Hom-objects in $Χ_{\mathcal{C}}(\mathcal{C}^{op})$.
• It uses the category of left $A$-modules in a monoidal category $\mathcal{C}$ as the primary setting, with duality and module category structures central to the construction.
• The framework incorporates biclosed categories and defines $\mathcal{Z}'_{\mathcal{C}}(\mathcal{C}^{op})$ as the category of symmetric 2-contratraces, generalizing SAYD contramodules.
• A key technique is the use of right adjoint functors to transfer structures between categories, particularly via $^*F$ for monoidal functors $F: \mathcal{C} \to \mathcal{D}$.

Experimental results

Research questions

• RQ1 How can cyclic cohomology with coefficients be redefined to achieve better invariance under Morita and 2-Morita equivalences?
• RQ2 In what sense does the new derived definition of cyclic cohomology agree with the classical one, and where do they differ?
• RQ3 Can a generalized Shapiro’s lemma be formulated for cyclic cohomology with coefficients in monoidal categories via 2-functoriality?
• RQ4 What is the categorical role of contramodules and symmetric 2-contratraces in the context of Hopf-cyclic cohomology?
• RQ5 How does the absence of a natural $S^2$-like autoequivalence affect the definition of SAYD modules and contramodules in general monoidal categories?

Key findings

• The derived definition of cyclic cohomology with coefficients agrees with the classical definition in monoidal categories with a projective unit, as shown in Proposition 4.7.
• The new definition exhibits Morita invariance for unital associative algebras in $\mathcal{C}$, as proven in Proposition 4.10.
• The paper proves 2-Morita invariance in a generalized sense: if two monoidal categories $\mathcal{C}$ and $\mathcal{D}$ are related by a 2-functor between their module 2-categories, then cyclic cohomology is preserved under suitable conditions.
• Theorem 4.31 establishes a general adjunction-type result for cyclic cohomology, valid even when $\mathcal{C}$ and \mathcal{D}$ are not 2-Morita equivalent, via an admissible bimodule $\mathcal{N}$.
• In the case of $\mathcal{C} = {}_H\mathcal{M}$, the new definition recovers Hopf-cyclic cohomology with coefficients, and for non-semisimple Hopf algebras, it differs from the old definition, as noted in Remark 4.23.
• The category $\mathcal{Z}'_{\mathcal{C}}(\mathcal{C}^{op})$ is identified as the category of symmetric 2-contratraces, which generalizes the notion of SAYD contramodules in the absence of a natural $S^2$-autoequivalence.