arxiv: 2605.06661 · v1 · submitted 2026-05-07 · ❄️ cond-mat.str-el · hep-th· math-ph· math.CT· math.MP· math.QA

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Pro-Tensor Network

Gen Yue , Ansi Bai , Linqian Wu , Tian Lan

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:29 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-thmath-phmath.CTmath.MPmath.QA

keywords pro-tensor networkcategorificationtensor networksmany-body physicsLevin-Wen modelstring-net modelsgeneralized symmetrytopological holography

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The pith

Pro-tensor networks categorify tensor networks to study many-many-body theories without semisimplicity or finiteness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the pro-tensor network as a categorification of the tensor network. This structure supplies a rigorous yet graphically transparent framework for analyzing collections of many-body theories, which the authors call many-many-body theory. It includes a toolbox for performing calculations that preserve graphical intuition. Applications recover the Levin-Wen model as a uniform pro-tensor network and extend results on particles by characterizing them as modules over promonads. The approach also interprets string-net pro-tensor networks as spaces of symmetric tensor networks, extending to generalized symmetry and topological holography while removing the usual requirements of semisimplicity, finiteness, and rigidity.

Core claim

The pro-tensor network is introduced as a categorification that represents many many-body theories in a fully rigorous manner while retaining graphical transparency for calculations. This recovers the Levin-Wen model as a uniform pro-tensor network and characterizes particles as modules over promonads, generalizing earlier work by Kitaev and Kong. The string-net version is interpreted as the space of symmetric tensor networks, enabling applications to generalized symmetry and topological holography. The construction operates without assuming semisimplicity, finiteness, or rigidity.

What carries the argument

The pro-tensor network, a categorification of the tensor network that encodes collections of many-body theories through promonads and graphical rules.

If this is right

The Levin-Wen model is recovered as a uniform pro-tensor network.
Particles are characterized as modules over promonads, generalizing the Kitaev-Kong result.
String-net pro-tensor networks correspond to the space of symmetric tensor networks.
The framework applies directly to studies of generalized symmetry and topological holography.
Graphical calculations remain valid even when semisimplicity, finiteness, or rigidity fail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This structure could support analysis of physical systems whose categories are infinite or non-semisimple, such as certain continuum limits in condensed matter.
It may link tensor-network techniques more closely with higher-categorical methods used in topological quantum field theory.
Numerical or analytic tests on models known to violate the relaxed assumptions could reveal new phases or symmetries.

Load-bearing premise

A well-defined pro-tensor network can be constructed and used for physical calculations while dispensing with the assumptions of semisimplicity, finiteness, and rigidity.

What would settle it

A concrete calculation on a many-body model using the pro-tensor network that produces results inconsistent with established physical predictions or experimental observations in that model would challenge the framework.

read the original abstract

We introduce the pro-tensor network, a categorification of the tensor network, as a fully rigorous yet graphically transparent framework for studying the collection of many many-body theories, which we dub many-many-body theory. We provide a comprehensive toolbox for the graphical calculations using pro-tensor networks. As applications, we recover the Levin-Wen model as a "uniform" pro-tensor network and generalize a result of Kitaev and Kong by characterizing particles as modules over promonads. One can also interpret the string-net pro-tensor network as the space of symmetric tensor networks, thus our framework also applies to the study of generalized symmetry and topological holography. Notably, our generalization dispenses with the assumptions of semisimplicity, finiteness, and rigidity, potentially facilitating the exploration of many-body physics beyond these constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces the pro-tensor network as a categorification of the tensor network, providing a rigorous yet graphically transparent framework for many-many-body theory. It supplies a toolbox of graphical rules, recovers the Levin-Wen model as a uniform pro-tensor network, generalizes Kitaev-Kong by characterizing particles as modules over promonads, and interprets the string-net pro-tensor network as the space of symmetric tensor networks. The framework is asserted to apply to generalized symmetry and topological holography while dispensing with the assumptions of semisimplicity, finiteness, and rigidity.

Significance. If the central constructions are shown to be well-defined and the graphical calculus remains coherent and physically usable, the work would represent a meaningful extension of tensor-network methods to categories lacking standard structural assumptions, potentially enabling new explorations in many-body physics, generalized symmetries, and holography. The emphasis on graphical transparency alongside categorification is a constructive feature.

major comments (2)

[Abstract and applications sections] The abstract asserts that the pro-tensor network and promonad constructions remain rigorous and yield transparent diagrams without semisimplicity, finiteness, or rigidity, yet the provided description indicates that explicit recoveries of Levin-Wen and Kitaev-Kong are demonstrated only in regimes satisfying those assumptions. A load-bearing gap is the absence of concrete verification that the module interpretations and symmetric-tensor-network reading continue to produce coherent fusion rules and physical predictions once those properties are removed.
[Graphical calculus toolbox] The claim that the framework dispenses with rigidity (no duals for wire bending) requires demonstration that the promonad graphical calculus preserves the necessary coherence for particle interpretations and string-net contractions; standard tensor-network and string-net calculi rely on rigidity for these operations, and the manuscript must show how the pro-construction substitutes for it without introducing inconsistencies.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript introducing pro-tensor networks. We address the major comments point by point below, providing clarifications on the generality of the framework and outlining the revisions we will make to strengthen the exposition.

read point-by-point responses

Referee: [Abstract and applications sections] The abstract asserts that the pro-tensor network and promonad constructions remain rigorous and yield transparent diagrams without semisimplicity, finiteness, or rigidity, yet the provided description indicates that explicit recoveries of Levin-Wen and Kitaev-Kong are demonstrated only in regimes satisfying those assumptions. A load-bearing gap is the absence of concrete verification that the module interpretations and symmetric-tensor-network reading continue to produce coherent fusion rules and physical predictions once those properties are removed.

Authors: The definitions of pro-tensor networks and promonads are formulated in complete generality using only the axioms of promonads in a 2-categorical setting, without any appeal to semisimplicity, finiteness, or rigidity. The recoveries of Levin-Wen and the generalization of Kitaev-Kong are presented as concrete illustrations in the standard semisimple setting to connect with existing literature, but the characterizations of particles as modules over promonads and the symmetric-tensor-network interpretation are proven using only the universal properties of promonads. Coherence of fusion rules follows from the associativity and unit axioms of the promonad, which hold independently of those assumptions. We will revise the abstract and applications sections to explicitly separate the general theorems from the illustrative examples and add a remark verifying that the module and string-net readings yield coherent predictions abstractly. This is a partial revision. revision: partial
Referee: [Graphical calculus toolbox] The claim that the framework dispenses with rigidity (no duals for wire bending) requires demonstration that the promonad graphical calculus preserves the necessary coherence for particle interpretations and string-net contractions; standard tensor-network and string-net calculi rely on rigidity for these operations, and the manuscript must show how the pro-construction substitutes for it without introducing inconsistencies.

Authors: In the pro-tensor network, operations such as wire bending and contractions are defined directly via the multiplication and unit of the promonad, which serve as substitutes for the rigidity isomorphisms of standard tensor networks. The graphical calculus inherits coherence from the coherence theorems for monads in 2-categories, ensuring consistency for module interpretations and string-net contractions. We agree that an explicit derivation of these substitution rules would make the toolbox more self-contained. We will add a dedicated subsection to the graphical calculus toolbox that derives the bending and contraction rules from promonad data alone and proves that no inconsistencies arise with the particle and string-net interpretations. This is a full revision to the relevant section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework is a self-contained categorification

full rationale

The paper constructs pro-tensor networks and promonads from standard category theory (promonads, modules, symmetric tensor networks) and derives graphical rules, Levin-Wen recovery, and Kitaev-Kong generalizations directly from these definitions. No equations or central claims reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations. The extension beyond semisimplicity/finiteness/rigidity is asserted as a consequence of the new objects rather than presupposed in their definition. The derivation chain remains independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the existence of a categorification that preserves graphical transparency while relaxing standard categorical assumptions; it draws on background category theory without introducing fitted numerical parameters.

axioms (1)

standard math Standard axioms of monoidal categories and graphical calculus
The framework relies on these to define pro-tensor networks and their calculations.

invented entities (2)

pro-tensor network no independent evidence
purpose: Categorification of tensor networks for many-many-body theories
Newly introduced object whose definition and properties form the core of the paper.
promonad no independent evidence
purpose: Algebraic structure used to characterize particles in the generalized setting
Extended or introduced to generalize the Kitaev-Kong result.

pith-pipeline@v0.9.0 · 5444 in / 1282 out tokens · 46475 ms · 2026-05-08T05:29:00.531655+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 18 canonical work pages · 1 internal anchor

[1]

Kong,Global quantum many-body theory, https://liang-kong.github.io/

L. Kong,Global quantum many-body theory, https://liang-kong.github.io/
[2]

Kong and H

L. Kong and H. Zheng,Gapless edges of 2d topological orders and enriched monoidal categories,Nuclear Physics B927(2018) 140–165

2018
[3]

Ji and X.-G

W. Ji and X.-G. Wen,Categorical symmetry and noninvertible anomaly in symmetry-breaking and topological phase transitions,Physical Review Research2(2020)

2020
[4]

Kong and H

L. Kong and H. Zheng,A mathematical theory of gapless edges of 2d topological orders. part i,Journal of High Energy Physics2020(2020)

2020
[5]

L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng,Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry,Physical Review Research2(2020)

2020
[6]

Chen, C.-M

W.-Q. Chen, C.-M. Jian, L. Kong, Y.-Z. You and H. Zheng,Topological phase transition on the edge of two-dimensional topological order,Physical Review B102(2020)

2020
[7]

L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang and H. Zheng,Classification of topological phases with finite internal symmetries in all dimensions,Journal of High Energy Physics2020 (2020) . – 92 –

2020
[8]

Lichtman, R

T. Lichtman, R. Thorngren, N.H. Lindner, A. Stern and E. Berg,Bulk anyons as edge symmetries: Boundary phase diagrams of topologically ordered states,Physical Review B104 (2021)

2021
[9]

Kong, X.-G

L. Kong, X.-G. Wen and H. Zheng,One dimensional gapped quantum phases and enriched fusion categories,Journal of High Energy Physics2022(2022)

2022
[10]

Kong and H

L. Kong and H. Zheng,Categories of quantum liquids i,Journal of High Energy Physics 2022(2022)

2022
[11]

Chatterjee and X.-G

A. Chatterjee and X.-G. Wen,Symmetry as a shadow of topological order and a derivation of topological holographic principle,Physical Review B107(2023)

2023
[12]

Moradi, S.F

H. Moradi, S.F. Moosavian and A. Tiwari,Topological holography: Towards a unification of landau and beyond-landau physics,SciPost Physics Core6(2023)

2023
[13]

Freed, G

D. Freed, G. Moore and C. Teleman,Topological symmetry in quantum field theory, Quantum Topology15(2024) 779

2024
[14]

Kong and H

L. Kong and H. Zheng,Categories of quantum liquids ii,Communications in Mathematical Physics405(2024)

2024
[15]

Lan and J.-R

T. Lan and J.-R. Zhou,Quantum current and holographic categorical symmetry,SciPost Physics16(2024)

2024
[16]

Xu and Z.-H

R. Xu and Z.-H. Zhang,Categorical descriptions of one-dimensional gapped phases with abelian onsite symmetries,Physical Review B110(2024)

2024
[17]

R. Wen, W. Ye and A.C. Potter,Topological holography for fermions,2404.19004

work page arXiv
[18]

Bhardwaj, L.E

L. Bhardwaj, L.E. Bottini, D. Pajer and S. Sch¨ afer-Nameki,Categorical landau paradigm for gapped phases,Phys. Rev. Lett.133(2024) 161601

2024
[19]

T. Lan, G. Yue and L. Wang,Category of set orders,Journal of High Energy Physics2024 (2024)

2024
[20]

Z. Jia, S. Tan and D. Kaszlikowski,Weak hopf symmetry and tube algebra of the generalized multifusion string-net model,Journal of High Energy Physics2024(2024)

2024
[21]

Cordova, D

C. Cordova, D. Garc´ ıa-Sep´ ulveda and N. Holfester,Particle-soliton degeneracies from spontaneously broken non-invertible symmetry,2403.08883

work page arXiv
[22]

Representation Theory of Solitons,

C. Cordova, N. Holfester and K. Ohmori,Representation theory of solitons,2408.11045

work page arXiv
[23]

Generalized Tube Algebras, Symmetry-Resolved Partition Functions, and Twisted Boundary States,

Y. Choi, B.C. Rayhaun and Y. Zheng,Generalized tube algebras, symmetry-resolved partition functions, and twisted boundary states,2409.02159

work page arXiv
[24]

Jones,DHR bimodules of quasi-local algebras and symmetric quantum cellular automata,2304.00068

C. Jones,Dhr bimodules of quasi-local algebras and symmetric quantum cellular automata, 2304.00068

work page arXiv
[25]

Lan,Tube category, tensor renormalization and topological holography,Communications in Mathematical Physics406(2025)

T. Lan,Tube category, tensor renormalization and topological holography,Communications in Mathematical Physics406(2025)

2025
[26]

Huang,Fermionic quantum criticality through the lens of topological holography,Phys

S.-J. Huang,Fermionic quantum criticality through the lens of topological holography,Phys. Rev. B111(2025) 155130

2025
[27]

An operator algebraic approach to fusion category symmetry on the lattice

D.E. Evans and C. Jones,An operator algebraic approach to fusion category symmetry on the lattice,2507.05185

work page internal anchor Pith review Pith/arXiv arXiv
[28]

Levin and X.-G

M.A. Levin and X.-G. Wen,String-net condensation:a physical mechanism for topological phases,Physical Review B71(2005) . – 93 –

2005
[29]

B´ enabou,Les distributeurs,Universit´ e Catholique de Louvain, Institut de Math´ ematique Pure et Appliqu´ eerapport 33(1973)

J. B´ enabou,Les distributeurs,Universit´ e Catholique de Louvain, Institut de Math´ ematique Pure et Appliqu´ eerapport 33(1973)

1973
[30]

Lawvere,Metric spaces, generalized logic, and closed categories,Rendiconti del Seminario Matematico e Fisico di Milano43(1973) 135–166

F.W. Lawvere,Metric spaces, generalized logic, and closed categories,Rendiconti del Seminario Matematico e Fisico di Milano43(1973) 135–166

1973
[31]

Kelly,Basic concepts of enriched category theory, London Mathematical Society lecture note series, Cambridge Univ

G.M. Kelly,Basic concepts of enriched category theory, London Mathematical Society lecture note series, Cambridge Univ. Pr, Cambridge (1982)

1982
[32]

Street,Enriched categories and cohomology,Quaestiones Mathematicae6(1983) 265–283

R. Street,Enriched categories and cohomology,Quaestiones Mathematicae6(1983) 265–283

1983
[33]

B´ enabou,Distributors at work,Lecture notes written by Thomas Streicher11(2000) 8

J. B´ enabou,Distributors at work,Lecture notes written by Thomas Streicher11(2000) 8

2000
[34]

Boisseau,Understanding profunctor optics: a representation theorem,2001.11816

G. Boisseau,Understanding profunctor optics: a representation theorem,2001.11816

work page arXiv 2001
[35]

Clarke, D

B. Clarke, D. Elkins, J. Gibbons, F. Loregian, B. Milewski, E. Pillmore et al.,Profunctor optics, a categorical update,Compositionality6(2024) 1

2024
[36]

Pastro and R

C. Pastro and R. Street,Doubles for monoidal categories,Theory and Applications of Categories21(2008) 61–75 [0711.1859]

work page arXiv 2008
[37]

Tambara,Distributors on a tensor category,Hokkaido Mathematical Journal35(2006) 379

D. Tambara,Distributors on a tensor category,Hokkaido Mathematical Journal35(2006) 379

2006
[38]

L´ opez Franco,Hopf modules for autonomous pseudomonoids and the monoidal centre, 0710.3853

I.L. L´ opez Franco,Hopf modules for autonomous pseudomonoids and the monoidal centre, 0710.3853

work page arXiv
[39]

Kitaev and L

A. Kitaev and L. Kong,Models for gapped boundaries and domain walls,Communications in Mathematical Physics313(2012) 351–373

2012
[40]

Kong,Some universal properties of Levin-Wen models, in17th International Congress on Mathematical Physics, pp

L. Kong,Some universal properties of Levin-Wen models, in17th International Congress on Mathematical Physics, pp. 444–455, 2013 [1211.4644]

work page arXiv 2013
[41]

Kong and X.-G

L. Kong and X.-G. Wen,Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions,1405.5858

work page arXiv
[42]

Johnson-Freyd,On the classification of topological orders,Communications in Mathematical Physics393(2022) 989–1033

T. Johnson-Freyd,On the classification of topological orders,Communications in Mathematical Physics393(2022) 989–1033

2022
[43]

Bai and Z.-H

A. Bai and Z.-H. Zhang,On the representation categories of weak hopf algebras arising from levin-wen models,2503.06731

work page arXiv
[44]

Kapranov and V

M. Kapranov and V. Voevodsky,2-categories and Zamolodchikov tetrahedra equations, in Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, vol. 56 ofProceedings of Symposia in Pure Mathematics, (Providence, RI), pp. 177–259, American Mathematical Society (1994)

1994
[45]

Baez and J

J.C. Baez and J. Dolan,Higher-dimensional algebra and topological quantum field theory, Journal of Mathematical Physics36(1995) 6073–6105

1995
[46]

Gaiotto and T

D. Gaiotto and T. Johnson-Freyd,Condensations in higher categories,
[47]

Day and R

B. Day and R. Street,Monoidal bicategories and hopf algebroids,Advances in Mathematics 129(1997) 99–157

1997
[48]

Joyal and R

A. Joyal and R. Street,The geometry of tensor calculus, i,Advances in Mathematics88 (1991) 55–112

1991
[49]

Justesen,Bikategorien af profunktorer, Master’s thesis, Aarhus University, Aarhus, 1968

M. Justesen,Bikategorien af profunktorer, Master’s thesis, Aarhus University, Aarhus, 1968. – 94 –

1968
[50]

Day,On closed categories of functors, inReports of the Midwest Category Seminar IV, S

B. Day,On closed categories of functors, inReports of the Midwest Category Seminar IV, S. MacLane, H. Applegate, M. Barr, B. Day, E. Dubuc, Phreilambud et al., eds., (Berlin, Heidelberg), pp. 1–38, Springer Berlin Heidelberg, 1970

1970
[51]

Street,Cauchy characterization of enriched categories,Rendiconti del Seminario Matematico e Fisico di Milano51(1981) 217–233

R. Street,Cauchy characterization of enriched categories,Rendiconti del Seminario Matematico e Fisico di Milano51(1981) 217–233

1981
[52]

Borceux and I

F. Borceux and I. Stubbe,Short introduction to enriched categories, inCurrent Research in Operational Quantum Logic: Algebras, Categories, Languages, B. Coecke, D. Moore and A. Wilce, eds., (Dordrecht), pp. 167–194, Springer Netherlands (2000), DOI

2000
[53]

Street,Frobenius monads and pseudomonoids,Journal of Mathematical Physics45(2004) 3930–3948

R. Street,Frobenius monads and pseudomonoids,Journal of Mathematical Physics45(2004) 3930–3948

2004
[54]

2022 , PAGES =

E. Riehl and D. Verity,Elements of∞-Category Theory, Cambridge University Press, 1 ed. (Jan., 2022), 10.1017/9781108936880

work page doi:10.1017/9781108936880 2022
[55]

Huang, H

M. Huang, H. Xu and Z.-H. Zhang,The 2-character theory for finite 2-groups, 2024

2024
[56]

Loregian,(Co)end Calculus, Cambridge University Press, 1 ed

F. Loregian,(Co)end Calculus, Cambridge University Press, 1 ed. (June, 2021), 10.1017/9781108778657

work page doi:10.1017/9781108778657 2021
[57]

Etingof, S

P.I. Etingof, S. Gelaki, D. Nikshych and V. Ostrik,Tensor categories, Mathematical surveys and monographs, American Mathematical Society, Providence, Rhode Island (2015)

2015
[58]

Marmolejo,Doctrines whose structure forms a fully faithful adjoint string,Theory and Applications of Categories3(1997)

F. Marmolejo,Doctrines whose structure forms a fully faithful adjoint string,Theory and Applications of Categories3(1997)

1997
[59]

L´ opez Franco, R

I. L´ opez Franco, R. Street and R.J. Wood,Duals invert,Applied Categorical Structures19 (2011) 321–361

2011
[60]

Majid,Anyonic quantum groups, inSpinors, Twistors, Clifford Algebras and Quantum Deformations, Z

S. Majid,Anyonic quantum groups, inSpinors, Twistors, Clifford Algebras and Quantum Deformations, Z. Oziewicz, B. Jancewicz and A. Borowiec, eds., (Dordrecht), pp. 327–336, Springer Netherlands, 1993

1993
[61]

Schomerus,Construction of field algebras with quantum symmetry from local observables, Communications in Mathematical Physics169(1995) 193–236

V. Schomerus,Construction of field algebras with quantum symmetry from local observables, Communications in Mathematical Physics169(1995) 193–236

1995
[62]

Nill and K

F. Nill and K. Szlach´ anyi,Quantum chains of hopf algebras with quantum double cosymmetry,Communications in Mathematical Physics187(1997) 159–200

1997
[63]

Connes and D

A. Connes and D. Kreimer,Hopf algebras, renormalization and noncommutative geometry, Communications in Mathematical Physics199(1998) 203–242

1998
[64]

Kitaev,Anyons in an exactly solved model and beyond,Annals of Physics321(2006) 2–111

A. Kitaev,Anyons in an exactly solved model and beyond,Annals of Physics321(2006) 2–111

2006
[65]

Chikhladze, S

D. Chikhladze, S. Lack and R. Street,Hopf monoidal comonads,Theory and Applications of Categories24(2010) 554–563

2010
[66]

Street,The formal theory of monads,Journal of Pure and Applied Algebra2(1972) 149–168

R. Street,The formal theory of monads,Journal of Pure and Applied Algebra2(1972) 149–168

1972
[67]

Willerton,A diagrammatic approach to hopf monads,

S. Willerton,A diagrammatic approach to hopf monads,
[68]

Lan and X.-G

T. Lan and X.-G. Wen,Topological quasiparticles and the holographic bulk-edge relation in (2+1)-dimensional string-net models,Physical Review B90(2014)

2014
[69]

Green, P

D. Green, P. Huston, K. Kawagoe, D. Penneys, A. Poudel and S. Sanford,Enriched string-net models and their excitations,Quantum8(2024) 1301. – 95 –

2024
[70]

C.-H. Lin, M. Levin and F.J. Burnell,Generalized string-net models: A thorough exposition, Physical Review B103(2021) [2012.14424]

work page arXiv 2021
[71]

Ocneanu,Chirality for operator algebras,Subfactors(1994) 39

A. Ocneanu,Chirality for operator algebras,Subfactors(1994) 39

1994
[72]

Izumi,The structure of sectors associated with Longo-Rehren inclusions

M. Izumi,The structure of sectors associated with Longo-Rehren inclusions. I: General theory,Commun. Math. Phys.213(2000) 127

2000
[73]

Mueger,From subfactors to categories and topology II: The quantum double of tensor categories and subfactors,J

M. Mueger,From subfactors to categories and topology II: The quantum double of tensor categories and subfactors,J. Pure Appl. Algebra180(2001) 159 [math/0111205]

work page arXiv 2001
[74]

Barter, J

D. Barter, J. Bridgeman and R. Wolf,Computing associators of endomorphism fusion categories,SciPost Physics13(2022) [2110.03644]

work page arXiv 2022
[75]

Konechny and V

A. Konechny and V. Vergioglou,On fusing matrices associated with conformal boundary conditions,2405.10189

work page arXiv
[76]

Kelly and V

G.M. Kelly and V. Schmitt,Notes on enriched categories with colimits of some class,Theory and Applications of Categories14(2005) 399–423. – 96 –

2005