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Action functor formalism

Given a monoidal category C = (C,⊗, 1), we denote its Drinfeld center by Z(C). If the forgetful functor U : Z(C) → C admits a right adjoint, say R, then the adjoint object of C is defined by AC := UR(1). Our main concern is the case where C is a finite tensor category in the sense of Etingof-Ostrik [1]. Some fundamental results on finite-dimensional Hopf algebras have been extended to the setting of finite tensor categories by using the adjoint object and the adjunction U ⊣ R [3, 4]. Here a naive question arises: Why is the adjoint object useful for this kind of problems? As the adjoint object is defined in terms of the tensor product of C, there is no obvious reason why it relates to somewhat ring-theoretic or representation-theoretic problems. In this talk, I introduce an abstract framework connecting the adjoint object and several ring-theoretic notions and review how results on Hopf algebras are extended to the setting of finite tensor categories. Let C be a finite tensor category. A key ingredient is the ‘action’ functor ρ : C → Rex(C) defined by ρ(X) = X⊗ (−), where Rex(C) is the category of right exact linear endofunctors on C. It turns out that ρ has a right adjoint, say ρ, and the adjoint object AC is isomorphic to ρ (idC). If we pick an arbitrary algebra L such that C ≈ L-mod, then Rex(C) ≈ L-bimod. Some ring-theoretic notions can be formulated in terms of the category of bimodules. If a ring-theoretic notion which we aim to investigate has such a description, then one can transport it to the category C through the equivalence L-bimod ≈ Rex(C) and the functor ρ : Rex(C) → C. This allows us to discuss relations between the notion and the adjoint object. As explained in [5], this formalism has a lot of applications. For example, ExtC(1, AC) is shown to be isomorphic to the Hochschild cohomology HH•(L). Noteworthy, this result extends the SL2(Z)-action on the Hochschild cohomology of a ribbon factorisable Hopf algebra to the setting of non-semisimple modular tensor categories. Under the assumption that the double dual functor on C is isomorphic to the identity functor, ExtC(AC, 1) is shown to be dual to Hochschild homology HH•(L) by a similar argument and an abstract treatment of the Nakayama functor established in [2]. Thus, under the same assumption, HomC(AC, 1) is isomorphic to the space of symmetric linear forms on L. I will show further applications of this kind of techniques. If time permits, I will talk about a generalization to modules over a finite tensor category.