(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

Abstract

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong $(\infty ,n)$-functors. We construct a double $(\infty ,n)$-category built out of the target $(\infty ,n)$-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty ,n)$-category $\mathcal{C}$ to an $(\infty ,n+d)$-category using the higher morphisms in $\mathcal{C}$.

arXiv comment58 pages. Many TikZ diagrams. Introduction has been rewritten. Most of Section 7 is new

References in corpus (20)

• Higher Topos Theory
• Higher-dimensional Algebra and Topological Quantum Field Theory
• Higher Algebraic Structures and Quantization
• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• A cartesian presentation of weak n-categories
• On the Classification of Topological Field Theories
• A Quillen model structure for Gray-categories
• Integrating quantum groups over surfaces
• Iterated spans and classical topological field theories
• On the Unicity of the Homotopy Theory of Higher Categories
• A note on the $(\infty ,n)$-category of cobordisms
• Perturbative algebraic quantum field theory
• The higher Morita category of $E_n$-algebras
• Rectification of enriched infinity-categories
• (Infinity,2)-Categories and the Goodwillie Calculus I
• Segal topoi and stacks over Segal categories
• Comparison of models for $(\infty ,n)$-categories, I
• Frobenius algebras and planar open string topological field theories
• Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors
• A model structure on internal categories in simplicial sets

Cited by in corpus (30)

• On the classification of topological orders
• Integrating quantum groups over surfaces
• Topological dualities in the Ising model
• Iterated spans and classical topological field theories
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
• A note on the $(\infty ,n)$-category of cobordisms
• The balanced tensor product of module categories
• The higher Morita category of $E_n$-algebras
• Categories of quantum liquids I
• Gapped boundary theories in three dimensions
• On dualizability of braided tensor categories
• Invertible braided tensor categories
• Composing topological domain walls and anyon mobility
• Comparison of models for $(\infty ,n)$-categories, II
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Condensations in higher categories
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• All about the Grothendieck construction
• Extended quantum field theory, index theory and the parity anomaly
• A 3-categorical perspective on G-crossed braided categories
• Fusion 3-Categories for Duality Defects
• On lax transformations, adjunctions, and monads in $(\infty ,2)$-categories
• Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories
• Compact Semisimple 2-Categories
• On the Dualizability of Fusion 2-Categories
• Coherence for adjunctions in a $3$-category via string diagrams
• Characters and transfer maps via categorified traces
• *-Autonomous Envelopes and Conservativity
• Orbifold completion of 3-categories
• Unit inclusion in a non-semisimple braided tensor category and non-compact relative TQFTs