[Paper Review] Further results on the structure of (co)ends in finite tensor categories

Kenichi Shimizu|arXiv (Cornell University)|Jan 8, 2018

Algebraic structures and combinatorial models31 references3 citations

TL;DR

This paper establishes a general framework for analyzing (co)ends in finite tensor categories by introducing a right adjoint functor ρ^ra to the action functor ρ: C → Rex(M), showing it is given by an end involving internal Hom functors. The key contribution is that Ext^•_C(1, ρ^ra(id_M)) is isomorphic to the Hochschild cohomology of M, enabling a projective action of the modular group SL₂(ℤ) on this cohomology in modular tensor categories.

ABSTRACT

Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. The action of $\mathcal{C}$ on $\mathcal{M}$ induces a functor $ρ: \mathcal{C} o \mathrm{Rex}(\mathcal{M})$, where $\mathrm{Rex}(\mathcal{M})$ is the category of $k$-linear right exact endofunctors on $\mathcal{M}$. Our key observation is that $ρ$ has a right adjoint $ρ^{\mathrm{ra}}$ given by the end $ρ^{\mathrm{ra}}(F) = \int_{M \in \mathcal{M}} \underline{\mathrm{Hom}}(M, M)$. As an application, we establish the following results: (1) We give a description of the composition of the induction functor $\mathcal{C}_{\mathcal{M}}^* o \mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*)$ and Schauenburg's equivalence $\mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*) \approx \mathcal{Z}(\mathcal{C})$. (2) We introduce the space $\mathrm{CF}(\mathcal{M})$ of `class functions' of $\mathcal{M}$ and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $\mathrm{CF}(\mathcal{M})$ and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $\mathrm{Ext}_{\mathcal{C}}^{\bullet}(1, ρ^{\mathrm{ra}}(\mathrm{id}_{\mathcal{M}}))$ is isomorphic to the Hochschild cohomology of $\mathcal{M}$. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.

Motivation & Objective

• To develop a general framework for analyzing (co)ends in finite tensor categories using categorical adjunctions.
• To describe the composition of the induction functor and Schauenburg’s equivalence in terms of the right adjoint ρ^ra.
• To initiate character theory for pivotal module categories via the space of class functions CF(M).
• To relate the filtration of CF(M) to ring-theoretic notions like the Reynolds ideal.
• To establish an isomorphism between Ext^•_C(1, ρ^ra(id_M)) and the Hochschild cohomology of M, and deduce modular group actions.

Proposed method

• Define the action functor ρ: C → Rex(M) for a finite tensor category C and an exact left C-module category M.
• Show that ρ admits a right adjoint ρ^ra, explicitly given by ρ^ra(F) = ∫_{M∈M} Hom(M, F(M)) for F ∈ Rex(M).
• Prove that ρ^ra is a k-linear, faithful, exact functor when M is indecomposable and exact.
• Equip ρ^ra with a monoidal and bimodule structure using the universal dinatural transformation of the end.
• Use the structure of ρ^ra to lift categorical constructions and relate them to known invariants like the end A and coend L.
• Apply the framework to derive isomorphisms between Ext groups and Hochschild cohomology, and to analyze the modular group action.

Experimental results

Research questions

• RQ1 How can the right adjoint of the action functor ρ: C → Rex(M) be explicitly described in terms of categorical ends?
• RQ2 What is the role of ρ^ra in describing the composition of the induction functor and Schauenburg’s equivalence in the center of dual categories?
• RQ3 How does the space of class functions CF(M) relate to the characters of simple objects in a pivotal module category?
• RQ4 What is the significance of the filtration on CF(M), and how does it relate to Ext^1(M,M) and the Reynolds ideal?
• RQ5 How does the isomorphism between Ext^•_C(1, ρ^ra(id_M)) and Hochschild cohomology of M lead to a projective action of SL₂(ℤ) on this cohomology?

Key findings

• The right adjoint ρ^ra(F) is explicitly given by the end ∫_{M∈M} Hom(M, F(M)), providing a concrete expression for this key functor.
• The composition of the induction functor C^*_M → Z(C^*_M) and Schauenburg’s equivalence Z(C^*_M) ≈ Z(C) is described explicitly using ρ^ra.
• The space of class functions CF(M) is introduced, and for pivotal M, CF_1(M) is spanned by characters of simple objects, while CF_2(M) fits into an exact sequence involving Ext^1(L,L) for simple L.
• The filtration on CF(M) is shown to relate to the Reynolds ideal and its generalizations, particularly in the pivotal case.
• Ext^•_C(1, ρ^ra(id_M)) is isomorphic to the Hochschild cohomology of M, and this isomorphism implies that the modular group SL₂(ℤ) acts projectively on this cohomology in modular tensor categories.