Built on

Metric spaces, generalized logic, and closed categories
F. William Lawvere · 1973
Earlier work this paper cites.
Basic concepts of enriched category theory
G. M. Kelly · 1982
Earlier work this paper cites.
Topological quantum field theories
M. Atiyah · 1988
Earlier work this paper cites.
Higher dimensional algebra and topological quantum field theory
J.C. Baez and J. Dolan · 1995
Earlier work this paper cites.
S S -structures for k k -linear categories and the definition of a modular functor
Ulrike Tillmann · 1998
Earlier work this paper cites.
Sur des notions de n n -catégorie et n n -groupoïde non strictes via des ensembles multi-simpliciaux
Zouhair Tamsamani · 1999
Earlier work this paper cites.
On fusion categories
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik · 2005
Earlier work this paper cites.
String net condensation: A physical mechanism for topological phases
Michael A. Levin and Xiao-Gang Wen · 2005
Earlier work this paper cites.
Topological entanglement entropy
Alexei Kitaev and John Preskill · 2006
Earlier work this paper cites.
Detecting topological order in a ground state wave function
Michael A. Levin and Xiao-Gang Wen · 2006
Earlier work this paper cites.
On the classification of topological field theories
Original
Jacob Lurie · 2009
Earlier work this paper cites.
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
Original
Xie Chen, Zheng Cheng Gu, and Xiao Gang Wen · 2010
Earlier work this paper cites.
Chiral topological insulators, superconductors, and other competing orders in three dimensions
Original
Pavan Hosur, Shinsei Ryu, and Ashvin Vishwanath · 2010
Earlier work this paper cites.

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2019

Condensations in higher categories

Gaiotto, Davide, Johnson-Freyd, Theo

Understand

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.

Built on

Metric spaces, generalized logic, and closed categories
F. William Lawvere · 1973
Earlier work this paper cites.
Basic concepts of enriched category theory
G. M. Kelly · 1982
Earlier work this paper cites.
Topological quantum field theories
M. Atiyah · 1988
Earlier work this paper cites.
Higher dimensional algebra and topological quantum field theory
J.C. Baez and J. Dolan · 1995
Earlier work this paper cites.
S S -structures for k k -linear categories and the definition of a modular functor
Ulrike Tillmann · 1998
Earlier work this paper cites.
Sur des notions de n n -catégorie et n n -groupoïde non strictes via des ensembles multi-simpliciaux
Zouhair Tamsamani · 1999
Earlier work this paper cites.
On fusion categories
Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik · 2005
Earlier work this paper cites.
String net condensation: A physical mechanism for topological phases
Michael A. Levin and Xiao-Gang Wen · 2005
Earlier work this paper cites.
Topological entanglement entropy
Alexei Kitaev and John Preskill · 2006
Earlier work this paper cites.
Detecting topological order in a ground state wave function
Michael A. Levin and Xiao-Gang Wen · 2006
Earlier work this paper cites.
On the classification of topological field theories
Original
Jacob Lurie · 2009
Earlier work this paper cites.
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
Original
Xie Chen, Zheng Cheng Gu, and Xiao Gang Wen · 2010
Earlier work this paper cites.
Chiral topological insulators, superconductors, and other competing orders in three dimensions
Original
Pavan Hosur, Shinsei Ryu, and Ashvin Vishwanath · 2010
Earlier work this paper cites.

Similar

Surface operators in 3d topological field theory and 2d rational conformal field theory
Original
Anton Kapustin and Natalia Saulina · 2010
Cited alongside, same era.
On the unicity of the homotopy theory of higher categories
Original
Clark Barwick and Christopher Schommer-Pries · 2011
Cited alongside, same era.
Higher categories, colimits, and the blob complex
Original
Scott Morrison and Kevin Walker · 2011
Cited alongside, same era.
The Classification of Two-Dimensional Extended Topological Field Theories
Original
Christopher J. Schommer-Pries · 2011
Cited alongside, same era.
Orbifold completion of defect bicategories
Original

Then

Symmetry and topology in quantum matter
A. Kitaev · 2015
Later among the works it cites.
A new kind of topological quantum order: A dimensional hierarchy of quasiparticles built from stationary excitations
Original
Sagar Vijay, Jeongwan Haah, and Liang Fu · 2015
Later among the works it cites.
(Op)lax natural transformations, twisted field theories, and “even higher” Morita categories
Original
Theo Johnson-Freyd and Claudia Scheimbauer · 2016
Later among the works it cites.
Homotopy coherent adjunctions and the formal theory of monads
Emily Riehl and Dominic Verity · 2016
Later among the works it cites.
Line and surface defects in Reshetikhin-Turaev TQFT
Original
Nils Carqueville, Ingo Runkel, and Gregor Schaumann · 2017
Later among the works it cites.