# The Relative Monoidal Center and Tensor Products of Monoidal Categories

## TL;DR

This paper introduces a theory of relative monoidal centers and tensor products for monoidal categories over braided categories, establishing Morita duality and providing concrete examples involving quantum groups and Yetter-Drinfeld modules.

## Contribution

It develops the concept of augmented monoidal categories, constructs relative tensor products and centers, and links these to quantum algebra examples, extending Morita duality in monoidal category theory.

## Key findings

- Relative tensor products exist under certain conditions.
- The relative monoidal center is equivalent to categories of Yetter-Drinfeld modules.
- Applications to quantum groups and braided bialgebras are demonstrated.

## 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.