Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Enriched monoidal categories I: centers

Liang Kong, Wei Yuan, Zhi-Hao Zhang, Hao Zheng

Abstract

This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.

Enriched monoidal categories I: centers

Abstract

This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the -center (-center or -center) of an enriched (monoidal or braided monoidal) category, and compute the centers explicitly when the enriched (braided monoidal or monoidal) categories are obtained from the canonical constructions. These centers have important applications in the mathematical theory of gapless boundaries of 2+1D topological orders and that of topological phase transitions in physics. They also play very important roles in the higher representation theory, which is the focus of the second work in the series.

Paper Structure

This paper contains 22 sections, 35 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. 2-categories
  3. Examples of 2-categories
  4. Centers in 2-categories
  5. Enriched categories
  6. Enriched categories, functors and natural transformations
  7. 2-categories of enriched categories
  8. Pushforward 2-functors
  9. Canonical construction
  10. Enriched monoidal categories
  11. Definitions and examples
  12. Canonical construction
  13. The category of enriched endo-functors
  14. The E0-centers of enriched categroies in ECat
  15. Enriched braided monoidal categories
  16. ...and 7 more sections

Key Result

Proposition 2.18

The $E_0$-center $\text{Z}_0(x)$ of $x \in \mathbf{C}$, if exists, admits an $E_1$-algebra structure.

Theorems & Definitions (174)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 164 more