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 $E_0$-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.

56 pages. published version

References in corpus (19)

• Boundary-bulk relation in topological orders
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Gapless edges of 2d topological orders and enriched monoidal categories
• Infinite loop spaces, and coherence for symmetric monoidal bicategories
• Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers
• Categories of quantum liquids I
• Cartesian Bicategories II
• The center functor is fully faithful
• One dimensional gapped quantum phases and enriched fusion categories
• Drinfeld center of enriched monoidal categories
• Higher-Dimensional Algebra I: Braided Monoidal 2-Categories
• Composing topological domain walls and anyon mobility
• Pointed Drinfeld center functor
• The monoidal centre as a limit
• Enriched string-net models and their excitations
• Quantum Current and Holographic Categorical Symmetry
• Categories of quantum liquids III
• Extended TQFT arising from enriched multi-fusion categories

Cited by in corpus (4)

• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Tube Category, Tensor Renormalization and Topological Holography
• Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly