A Characterization of Braided Enriched Monoidal Categories

TL;DR

This paper establishes an equivalence between categories of braided V-monoidal categories and certain functor categories into Drinfeld centers, providing a new perspective on their structure and extensions.

Contribution

It constructs an equivalence between VMonCat and VModTens, linking braided V-monoidal categories with oplax braided functors into Drinfeld centers, including G-graded analogues.

Findings

Established an equivalence between VMonCat and VModTens categories.

Extended the equivalence to G-graded analogues and extensions.

Provided a framework for understanding V-monoidal categories via functor categories.

Abstract

We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The 1-cells in each are the respective lax monoidal functors, and the 2-cells are the respective monoidal natural transformations. Our proof also gives an equivalence in the case that we consider only strong monoidal 1-cells on both sides. The 2-categories VMonCat and VModTens have G-graded analogues. We also get an equivalence of 2-categories between G-extensions of some fixed V-monoidal category A, and G-extensions of some fixed V-module tensor category (A, F_A^Z).