Condensations in higher categories

Abstract

We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions.

37 pages. v3 corrects a nontrivial error

References in corpus (20)

• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• String-net condensation: A physical mechanism for topological phases
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Boson topological insulators: A window into highly entangled quantum phases
• Anyon condensation and its applications
• The Classification of Two-Dimensional Extended Topological Field Theories
• Chiral Topological Insulators, Superconductors and other competing orders in three dimensions
• Enriched $\infty$-categories via non-symmetric $\infty$-operads
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• On the Classification of Topological Field Theories
• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• Modular categories as representations of the 3-dimensional bordism 2-category
• A note on the $(\infty ,n)$-category of cobordisms
• Orbifolds of n-dimensional defect TQFTs
• Line and surface defects in Reshetikhin-Turaev TQFT
• Orbifolds of Reshetikhin-Turaev TQFTs
• Higher categories, colimits, and the blob complex
• A direct proof that the category of 3-computads is not cartesian closed

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• Classification of topological phases with finite internal symmetries in all dimensions
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Minimal nondegenerate extensions
• Symmetries and Anomalies of (1+1)d Theories: 2-groups and Symmetry Fractionalization
• One dimensional gapped quantum phases and enriched fusion categories
• Q-system completion for C* 2-categories
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• Finite Semisimple Module 2-Categories
• The 2-Deligne Tensor Product
• Matching higher symmetries across Intriligator-Seiberg duality
• Compact Semisimple 2-Categories
• Q-system completion is a 3-functor
• A universal state sum