The relative monoidal center and tensor products of monoidal categories

Abstract

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. The two main constructions are a relative tensor product of monoidal categories as well as a relative version of the monoidal center, which are Morita dual constructions. A general existence statement for a relative tensor products is derived from the existence of pseudo-colimits.In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result holds for small quantum groups, without restricting to locally finite weight modules. More generally, for modules over braided bialgebras, the relative center is shown to be equivalent to the category of braided Yetter-Drinfeld modules (or crossed modules). This category corresponds to modules over the braided Drinfeld double (or double bosonization) which are locally finite for the action of the dual.1.3. Summary of Results. In the spirit of moving from the representation theory of algebras to modules of categories (in this case, monoidal categories), this paper studies the representation theory of the relative monoidal center in the framework of [EGNO15]. After establishing the categorical setup and recalling generalities on categorical modules (Sections 2.1 & 3.1), the relative tensor product of categorical bimodules is reviewed in the generality of finitely cocomplete k-linear categories (Section 3.2 & Appendix A).The concept of the relative monoidal center from [Lau15] is refined using the language of Bbalanced functors to allow a better study of its categorical modules. For this, B-augmented monoidal categories are introduced in Section 3.3. These can be thought of as categorical analogues of Caugmented C-algebras R over a commutative k-algebra C. Over a B-augmented monoidal category, we can study B-balanced bimodules (Section 3.4). This construction is a categorical analogue of R b C R op -modules, over a C-algebra R. In other words, R-bimodules for which the left and right C-action coincide. Based on this concept, we present two constructions for a B-augmented monoidal category C:(1) the relative monoidal center Z B pCq, which is defined as Z B pCq " Hom B C-C pC, Cq, i.e. the category of endofunctors of B-balanced bimodule functors of the regular C-bimodule (Section 3.5);(2) the relative tensor product C B C op (Theorem 3.21) which is a monoidal category, generalizing the tensor product of categories of Kelly [Kel05]. The monoidal category C B C op has the universal property that its categorical modules give all B-balanced bimodules over C (Theorem 3.27). We show that there is a natural construction to turn a C-bimodule into a B-balanced C-bimodule.Theorem 3.42. Restriction along the monoidal functor C C op Ñ C B C op has a left 2-adjointThis construction is a categorical analogue of restricting a R-R-bimodule to the subspa...

Braided Commutative Algebras Over Quantized Enveloping Algebras

“…We present the main results of our work in this section. First, we discuss background material on B-central monoidal categories and relative monoidal centers Z B (C) adapted from [30]…”

“…The definition above generalizes the concept of a B-augmented monoidal category of [30,Sect. 3.3] relaxing the condition that the functor T has a left inverse.…”

“…) in Z(C). In [30], the relative monoidal center Z B (C) is defined (equivalently) as the monoidal category of B-balanced endofunctors G of the regular C-bimodule category C; here, using composition as the tensor product, i.e., G ⊗ G = G G, this is also a braided monoidal category.…”

“…Relative monoidal categories have the following properties, some of which hold by definition. Here, we use [30,Prop. 3.34] to employ other results from [30].…”

“…In this work, we build commutative algebras in relative monoidal centers Z B (C) [Definition 3.3], which is a class of braided monoidal categories studied by the first author [28], [30] (motivated by [2], [36], and related to [39,Def. 2.6]; see also [14,Sect.…”