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Drinfeld center of enriched monoidal categories

Liang Kong, Hao Zheng

TL;DR

The paper extends the Drinfeld center to monoidal categories enriched over a braided monoidal base, proving that the center Z(C^sharp) is itself an enriched braided monoidal category. It shows that every modular tensor category can be realized canonically as the center of a self-enriched monoidal category, and it provides a generalization linking centralizers and Müger centers with potential physics applications, including 2+1D TQFT. A constructive framework is developed for obtaining enriched monoidal structures from pairs (B,C) with braided oplax functors, clarifying how centers arise as centralizers within ordinary centers. These results unify enriched category theory with center constructions and offer a pathway to connect mathematical structures to extended topological quantum field theories.

Abstract

We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.

Drinfeld center of enriched monoidal categories

TL;DR

The paper extends the Drinfeld center to monoidal categories enriched over a braided monoidal base, proving that the center Z(C^sharp) is itself an enriched braided monoidal category. It shows that every modular tensor category can be realized canonically as the center of a self-enriched monoidal category, and it provides a generalization linking centralizers and Müger centers with potential physics applications, including 2+1D TQFT. A constructive framework is developed for obtaining enriched monoidal structures from pairs (B,C) with braided oplax functors, clarifying how centers arise as centralizers within ordinary centers. These results unify enriched category theory with center constructions and offer a pathway to connect mathematical structures to extended topological quantum field theories.

Abstract

We define the Drinfeld center of a monoidal category enriched over a braided monoidal category, and show that every modular tensor category can be realized in a canonical way as the Drinfeld center of a self-enriched monoidal category. We also give a generalization of this result for important applications in physics.

Paper Structure

This paper contains 5 sections, 8 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Enriched (monoidal) categories
  3. Enriched braided/symmetric monoidal categories
  4. Drinfeld center
  5. A generalization

Key Result

Proposition 2.4

Let $\mathcal{C}^\sharp$ be an enriched monoidal category. There is an enriched monoidal equivalence $\mathcal{C}^\sharp\simeq\mathcal{D}^\sharp$ where $\mathcal{D}^\sharp$ is a strict enriched monoidal category.

Theorems & Definitions (31)

  • Example 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Remark 3.3
  • ...and 21 more