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REVIEW 4 major objections 4 minor 135 references

Categorical quantum symmetries and ribbon tensor 2-categories

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Pith's one-line read The 2-representation 2-category of $\mathbb{U}_q\mathfrak{G}$ is a ribbon tensor 2-category, with constructed balancings and coherence data that yield framed 2-tangle invariants and a 4d 2-Chern-Simons TQFT.

desk verdict Genuinely useful higher-categorical framework, but the ribbon tensor 2-category claim is conditional on unproven quasi-Hermiticity assumptions and an imposed swallowking equation; the abstract overstates what is proven. read the letter →

arxiv 2501.08041 v2 pith:2KRFEB5W submitted 2025-01-14 math-ph math.CTmath.MPmath.QA

Hank Chen This is my paper · ORCID
classification math-phmath.CTmath.MPmath.QA MSC 18M1518N1018M2081T4581R50
keywords ribbontensor2-category2-representationHopfcategory2-Chern-Simonstheoryquantum2-gaugesymmetryframed2-tanglespivotalquasi-Hermitian2-R-matrix
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove that the 2-Hilb-enriched 2-category $\operatorname{2Rep}(\tilde C;\tilde R)$ of finite semisimple linear 2-representations of the categorical quantum symmetries $\tilde C = \mathbb{U}_q\mathfrak{G}$ is a ribbon tensor 2-category: braided, planar-pivotal, and lax rigid, with ribbon-balancing functors constructed explicitly. The reason to care is that such a structure is exactly the input needed to lift the classical ribbon-category invariant construction one categorical level, replacing decorated ribbon graphs by decorated 2-ribbons. If the proof is right, ribbon 2-functors from the standard 2-tangle 2-category into $\operatorname{2Rep}(\tilde C;\tilde R)$ produce framed invariants of 2-tangles and, via the 2-tangle hypothesis, a functorial 4d 2-Chern-Simons TQFT. The paper also shows that in the classical limit $q\to 1$ the same 2-category becomes symmetric and strict pivotal.

What carries the argument

The load-bearing machinery is the quasi-Hermitian 2-R-matrix $\tilde R$ on the Hopf category $\tilde C$. Condition (5.4), that the orientation-reversed element $\bar\nu$ coincides with the transposed element $\mu^T$, makes the braiding on dual objects self-adjoint and yields the writhing 2-morphisms from which the ribbon balancings $\vartheta_D,\bar\vartheta_D$ are assembled through the duality folds $\mathrm{ev}_D,\mathrm{cev}_D$. A second condition, the swallowking equations (5.12), glues the two swallowtail 2-morphisms together; this is what enforces the double-twist cancellations and the pivotality-type coherence needed for the ribbon structure.

What would settle it

Compute $\tilde R$ from the lattice quantization of the companion paper in the single-loop case and test condition (5.4): a single object with $\bar\nu \neq \mu^T$ invalidates the writhing identities and the ribbon balancing. A second direct check is whether the 2-Drinfel'd modifications $\omega_D,\bar\omega_D$ are invertible for every object $D$ and whether the swallowking equations (5.12) hold; a counterexample would break both the ribbon structure and the proof of classical pivotality.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is that the categorical quantum symmetries of 4d 2-Chern-Simons theory are rigid enough to support a full ribbon tensor 2-category structure, not merely a braiding. Starting from the cobraided Hopf dagger category $\tilde C=\mathbb{U}_q\mathfrak{G}$ and its 2-R-matrix $\tilde R$, the author constructs the ribbon balancings $\vartheta_D:{}^\ast D\to D^\ast$ and $\bar\vartheta_D:D^\ast\to{}^\ast D$ as composites of braiding maps on dual objects with the duality folds, and exhibits invertible 2-Drinfel'd modifications $\omega_D:\bar\vartheta_D^\ast\to\vartheta_{D^\ast}$ relating them. The coherence story is carried by writhing 2-morphisms, belt-buckle moves, and two types of double-twist cancellation, capped by 'swallowking' equations that paste the two swallowtail identities together. The outcome is a four-tier classification of framed-ness for objects (fully-framed, half-framed, unframed, self-dual) that specializes to earlier frameworks, and in the classical limit every object becomes unframed as the 2-category becomes symmetric and pivotal.

Load-bearing premise

The proof rests on the assumption, stated but not derived from the companion quantization, that the 2-R-matrix $\tilde R$ is quasi-Hermitian and that the swallowking equations hold; if either of these fails, the ribbon balancings and the pivotal classical limit collapse.

Editorial extensions

If this is right

  • Ribbon 2-functors from the 2-tangle 2-category into $\operatorname{2Rep}(\tilde C;\tilde R)$ produce framed invariants of 2-tangles, the 2-categorical analogue of decorated ribbon graphs.
  • By the 2-tangle hypothesis, those invariants assemble into a functorial 4d 2-Chern-Simons TQFT, with the ribbon balancing supplying the framing data the theory requires.
  • The four levels of framed-ness (fully-framed, half-framed, unframed, self-dual) unify and refine the earlier notions of framing in 2-categories with duals.
  • In the classical limit $q\to1$, the 2-category becomes symmetric and strict pivotal, recovering the usual pivotal structure for finite 2-group representations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if quasi-Hermiticity holds, the same 2-category offers a direct algebraic test of the conjectured equivalence between $\operatorname{2Rep}(\mathbb{U}_q\,\mathrm{inn}\,\mathfrak{sl}_2)$ and the condensation completion of the representation category of $\mathbb{U}_q\mathfrak{sl}_2$.
  • Editorial extension: the construction is probably not special to $\mathbb{U}_q\mathfrak{G}$; any cobraided Hopf dagger category with a quasi-Hermitian 2-R-matrix and valid swallowking equations should yield a ribbon tensor 2-category by the same argument.
  • Editorial extension: a concrete testable extension is to compute the ribbon balancings and Hopf-link orders in a finite 2-group or other explicitly known example; the order at which Hopf links become self-adjoint should track the order of $q$ at roots of unity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper claims to prove that for the Hopf category \tilde C = U_qG constructed in the author's companion paper [36], the 2-Hilb-enriched 2-category 2Rep(\tilde C;\tilde R) of finite semisimple C-linear module categories is braided, planar-pivotal, and lax rigid, and hence is a ribbon tensor 2-category. It constructs ribbon balancing functors \vartheta_D and \bar\vartheta_D, 2-Drinfeld modifications \omega_D and \bar\omega_D, and a hierarchy of framing notions (fully-framed, half-framed, unframed, self-dual), and uses the Baez-Langford 2-tangle hypothesis to motivate ribbon 2-functors and framed 2-tangle invariants. It also claims that in the classical limit q\to 1 the 2-category becomes strict pivotal in the sense of Douglas-Reutter. Most of the coherence constructions in Sections 5-6 presuppose quasi-Hermiticity (5.4) and the swallowking equations (5.12), which are assumed rather than derived for the specific input.

Significance. If the announced theorems were fully proven, the paper would provide a nontrivial 2-categorical analogue of a ribbon category and a concrete candidate input for Baez-Langford 2-tangle invariants and 4d 2-Chern-Simons TQFTs; the framing hierarchy (Tables 1 and 2) is a useful organizational contribution. The author is explicit about several technical assumptions, and the conditional structure of the construction is a strength in presentation. However, the central example-dependent claims are not established: the ribbon structure depends on quasi-Hermiticity (5.4), the classical-limit pivotality depends on the swallowking equations (5.12), and the framing hierarchy depends on invertibility of \omega_D and \bar\omega_D, none of which is proved here. The paper also relies substantially on the author's companion papers [36] and [50] for the input structures, so the announced example remains conditional on unverified inputs.

major comments (4)
  1. [§5.2.2, Eq. (5.4)] The ribbon statement in the abstract is not supported, because the condition named in Proposition 5.2 is assumed rather than verified. The text after Proposition 5.2 states that \tilde R will 'often' be assumed quasi-Hermitian, and Sections 6.1.2 and 6.1.3 use (5.4) to obtain the isomorphisms c_{D^*,D} \cong c^*_{D,D^*}, the belt-buckle moves (Proposition 6.3), the double-twist cancellations (Proposition 6.4), and the Reidemeister-I witnesses R_D and L_D. No proof is given that the 2-R-matrix \tilde R of Theorem 2.3 / [36] satisfies \bar\nu = \mu^T and \bar\mu = \nu^T; Remark 5.4 notes that nothing in the gauge theory forces \tilde C to be braided, so the condition is not automatic. The main theorem must either be explicitly conditioned on quasi-Hermiticity or provide a derivation of (5.4) for the constructed \tilde R.
  2. [§5.3.2, Eq. (5.12)] The swallowking equations are imposed as an additional condition, not derived, and they are load-bearing. Proposition 6.7 uses (5.12) in proving self-adjointness of the Hopf-link modification \Lambda_D, and Theorem 7.1 uses it to recover the C1 swallowtail equations in the classical limit. The proof of Theorem 7.1 only shows that, granted (5.12), the equation S_D \circ id_{flip} \circ S'_D = id_{flip} forces the swallowtails to be trivial; it does not prove (5.12) for q=1. Thus the classical-limit pivotality theorem is conditional on an unproved coherence identity.
  3. [§6.1.1, Definition 6.5] The framing hierarchy assumes the 2-Drinfeld modifications \omega_D and \bar\omega_D are invertible, but Proposition 6.1 only constructs these 2-morphisms; invertibility is not proved. Corollary 6.2 and the reductions in Table 2 rely on this invertibility, and Definition 6.5 makes it a hypothesis. Without an invertibility proof, the statement that 2Rep(\tilde C;\tilde R) carries the advertised ribbon-balancing structure is not established.
  4. [§1.1 and §6] The paper uses 'ribbon tensor 2-category' in the title and abstract as if it were a defined object, but no definition is given that lists the full coherence data. Definition 6.5 only defines framing refinements after assuming balancings, and the synoptic diagram in §1.1 is informal. The central claim that 2Rep(\tilde C;\tilde R) is a ribbon tensor 2-category therefore cannot be checked against a precise standard; a formal definition should appear before the main theorem.
minor comments (4)
  1. [§7, Theorem 7.1 proof] The proof refers to 'Remark 7', but no Remark 7 exists; the intended reference is likely Remark 3.5 about the unipotence of the antipode in the classical limit.
  2. [§5.3.2 and §8] After Eq. (5.12), the cross-reference '§6.11' should be corrected to a real section number, and the Conclusions refer to 'Remark 6.7' although the last remark before the Conclusions is Remark 6.6.
  3. [Throughout] There are typographical slips that should be fixed: 'quasi-Hemritian' in Proposition 6.3, 'Reidemsiter' in §6.1.3, and 'Kaufmann' where 'Kauffman' is meant.
  4. [§3.1.1, Remark 3.1] The standing assumption that all functors between monoidal products can be written using nudging is not mentioned in the statement of Theorem 3.1 or in the abstract; if this assumption is needed for the tensor product and braiding constructions, it should appear explicitly in the main theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ribbon conclusions are conditional on explicitly stated quasi-Hermiticity and invertibility assumptions, and the self-citations are to prior constructions rather than restatements of the target result.

full rationale

The paper's derivation chain is conditional rather than circular. The braided monoidal structure of 2Rep(\tilde C;\tilde R) is imported via Theorem 3.1 from the author's prior work [50] together with Neuchl [47]; this is a self-citation, but it supplies an independent prior construction of braided structures for Hopf 2-algebras and does not restate the ribbon claim made in this paper. The ribbon balancings \vartheta_D and \bar\vartheta_D, the 2-Drinfel'd modifications \omega_D and \bar\omega_D, the belt-buckle moves, and the Reidemeister II and I witnesses are all constructed in Sections 6.1.1--6.1.3 under the explicitly stated quasi-Hermitian hypothesis (5.4), and Section 6.1.3 additionally assumes invertibility of the fold-crossings and writhes. These are sufficient hypotheses that are assumed, not derived from an earlier theorem, and Remark 5.4 even notes that nothing in the gauge theory forces \tilde C to be braided. Thus the unproved gap is that the abstract states the ribbon conclusion unconditionally while the body establishes it only under (5.4); this is a completeness or correctness concern, not a case where the conclusion is definitionally identical to the inputs. Similarly, the swallowking equations (5.12) are imposed as conditions and then used in Theorem 7.1, where they become trivial in the classical limit; this is conditional derivation, not circularity. Definition 6.5 (half-framed, unframed, self-dual) is a taxonomy of framing notions rather than a source of the constructed structures, and no external dataset is fitted and then renamed as a prediction. No uniqueness theorem is imported from the authors' own work to force a choice, and no equation is quoted that reduces the central ribbon claim to its own input by construction. Hence the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 4 invented entities

There are no numerically fitted parameters in this pure mathematics paper; the formal deformation parameter q is an input, not a fitted constant. The load-bearing assumptions are structural: the existence and properties of the Hopf category U_q G come from an unpublished companion, quasi-Hermiticity and the swallowking equations are imposed rather than proven, and several coherence conventions such as nudging and planar-unitarity are assumed globally.

assumptions (6)
  • domain assumption Companion construction of U_q G as a strictly cobraided Hopf dagger category (Proposition 2.2, Theorem 2.3).
    Imported from the author's unpublished companion paper arXiv:2501.06486; everything downstream assumes it.
  • ad hoc to paper The 2-R-matrix \tilde R is quasi-Hermitian, equation (5.4).
    Assumed after Proposition 5.2 and used throughout Sections 5 and 6; not derived from the 2-Chern-Simons quantization.
  • ad hoc to paper Swallowking equations (5.12) hold.
    Imposed in Section 5.3.2 as a pasting of swallowtails and writhes; used to prove Proposition 6.7 and Theorem 7.1.
  • domain assumption Planar-unitarity of all structural functors, folds, snakerators, writhes and fold-crossings.
    Assumed for all 2-representations in Section 3.2.1 and for H, G, K in Section 6.1.3.
  • ad hoc to paper Nudging condition: all functors between monoidal products can be written via interchangers.
    Remark 3.1 states a 2-categorical analogue of Deligne's condition, not proven for 2Rep.
  • ad hoc to paper Invertibility of the 2-Drinfeld modifications \omega_D and \bar\omega_D.
    Constructed in Proposition 6.1 but used as a hypothesis in Definition 6.5 and Corollary 6.2.
invented entities (4)
  • Ribbon tensor 2-category
    purpose: Algebraic structure intended to model framed 2-tangles and provide ribbon 2-functors to a 4d TQFT.
    New notion proposed in Section 6; no external verification is available.
  • Ribbon balancings \vartheta_D and \bar\vartheta_D
    purpose: Encode the framing twist of objects in 2Rep.
    Constructed from braiding and duality data; internal to the paper.
  • 2-Drinfeld modifications \omega_D and \bar\omega_D
    purpose: Relate left-over and right-under ribbon balancings.
    Introduced in Proposition 6.1; invertibility is an assumption for framing definitions.
  • Swallowking equations
    purpose: Coherence condition pasting swallowtail diagrams with writhes.
    Imposed rather than derived; load-bearing for pivotality in Section 7.

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Cite this review

Pith. "Pith review of Categorical quantum symmetries and ribbon tensor 2-categories." pith.science (2026). https://pith.science/paper/2KRFEB5W

@misc{pith2026250108041,
  author       = {Pith},
  title        = {Pith review of: Categorical quantum symmetries and ribbon tensor 2-categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2KRFEB5W}},
  note         = {Machine review of arXiv:2501.08041}
}
abstract

In a companion work on the combinatorial quantization of 4d 2-Chern-Simons theory, the author has constructed the Hopf category of quantum 2-gauge transformations $\tilde{C}=\mathbb{U}_q\mathfrak{G}$ acting on the discrete surface-holonomy configurations on a lattice. We prove in this article that the 2-$\mathsf{Hilb}$-enriched 2-representation 2-category $\operatorname{2Rep}(\tilde C)$ of finite semisimple $\mathbb{C}$-linear $\tilde C$-module categories is braided, planar-pivotal, and lax rigid, hence $\operatorname{2Rep}(\tilde C)$ provides an example of a ribbon tensor 2-category. We explicitly construct the ribbon balancing functors, and exhibit their coherence conditions against the rigid dagger structures. This allows one to refine the various notions of \textit{framing} in a 2-category with duals that have been previously studied in the literature. Following the 2-tangle hypothesis of Baez-Langford, framed invariants of 2-tangles can then be constructed from ribbon 2-functors into $\operatorname{2Rep}(\tilde C)$, analogous to the definition of decorated ribbon graphs in the Reshetikhin-Turaev construction. We will also prove that, in the classical limit $q\rightarrow 1$, the 2-category $\operatorname{2Rep}(\mathbb{U}_{q=1}\mathfrak{G})$ becomes strict pivotal in the sense of Douglas-Reutter.

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Reference graph

Works this paper leans on

135 extracted references · 32 canonical work pages

  1. [36]

    Combinatorial quantization of 4d 2-Chern-Simons theory I: the Hopf category of higher-graph states

    Hank Chen. Combinatorial quantization of 4d 2-Chern-Simons theory I: the Hopf category of higher-graph states. 2025. arXiv:2501.06486 [math-ph]. url: https://arxiv.org/abs/2501. 06486

    arXiv 2025

  2. [42]

    Fusion 2-categories and a state-sum invariant for 4-manifolds

    Christopher L. Douglas and David J. Reutter. “Fusion 2-categories and a state-sum invariant for 4-manifolds”. In: arXiv: Quantum Algebra(2018). url: https://api.semanticscholar.org/ CorpusID:119305837

    2018

  3. [50]

    CategorifiedQuantumGroupsandBraidedMonoidal2-Categories

    HankChenandFlorianGirelli.“CategorifiedQuantumGroupsandBraidedMonoidal2-Categories”. In: (Apr. 2023). arXiv:2304.07398 [math.QA]

    work page arXiv 2023

  4. [1]

    Knot Theory and Statistical Mechanics

    Vaughan F. R. Jones. “Knot Theory and Statistical Mechanics”. In: Scientific American 263.5 (1990), pp. 98–105. issn: 00368733, 19467087. url: http://www.jstor.org/stable/24996978 (visited on 03/24/2024)

    arXiv 1990

  5. [2]

    Global aspects of current algebra

    Edward Witten. “Global aspects of current algebra”. In:Nuclear Physics B223.2 (1983), pp. 422–

    1983

  6. [3]

    Foundations of quantum group theory.CambridgeUniversityPress,2011

    S.Majid. Foundations of quantum group theory.CambridgeUniversityPress,2011. isbn:9780511834530, 9780521648684

    2011

  7. [4]

    Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU (N) groups

    S.L. Woronowicz. “Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU (N) groups.” In: Inventiones mathematicae 93.1 (1988), pp. 35–76. url: http://eudml.org/doc/ 143589

    1988

  8. [5]

    Quantum groups

    V. G. Drinfeld. “Quantum groups”. In: Zap. Nauchn. Semin. 155 (1986), pp. 18–49. doi: 10 . 1007/BF01247086

    1986

Show all 135 references
  1. [6]

    A q difference analog of U(g) and the Yang-Baxter equation

    Michio Jimbo. “A q difference analog of U(g) and the Yang-Baxter equation”. In: Lett. Math. Phys. 10 (1985), pp. 63–69.doi: 10.1007/BF00704588

    1985 doi

  2. [7]

    Hidden quantum groups inside Kac- Moody algebra

    A. Alekseev, L. D. Faddeev, and M. Semenov-Tian-Shansky. “Hidden quantum groups inside Kac- Moody algebra”. In:Commun. Math. Phys.149 (1992), pp. 335–345.doi: 10.1007/BF02097628

    1992 doi

  3. [8]

    Tensor Structures Arising from Affine Lie Algebras. III

    D. Kazhdan and G. Lusztig. “Tensor Structures Arising from Affine Lie Algebras. III”. In:Journal of the American Mathematical Society7.2 (1994), pp. 335–381.issn: 08940347, 10886834. url: http://www.jstor.org/stable/2152762 (visited on 11/18/2023)

    1994

  4. [9]

    Ribbon graphs and their invariants derived from quantum groups

    N. Yu. Reshetikhin and V. G. Turaev. “Ribbon graphs and their invariants derived from quantum groups”. In:Commun. Math. Phys.127 (1990), pp. 1–26.doi: 10.1007/BF02096491

    1990 doi

  5. [10]

    MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS

    VLADIMIR G. TURAEV. “MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS”. In: International Journal of Modern Physics B06.11n12 (1992), pp. 1807–1824.doi: 10.1142/ S0217979292000876. eprint: https : / / doi . org / 10 . 1142 / S0217979292000876. url: https : //doi.org/10.1142/S02179...

    1992 doi

  6. [11]

    Kitaev’s Lattice Model and Turaev-Viro TQFTs

    Benjamin Balsam and Alexander Kirillov Jr. Kitaev’s Lattice Model and Turaev-Viro TQFTs

  7. [12]

    Invariants of piecewise linear three manifolds

    John W. Barrett and Bruce W. Westbury. “Invariants of piecewise linear three manifolds”. In: Trans. Am. Math. Soc.348 (1996), pp. 3997–4022. doi: 10.1090/S0002- 9947- 96- 01660- 1 . arXiv: hep-th/9311155

    1996 arXiv

  8. [13]

    Reshetikhin

    N. Reshetikhin. Lectures on the integrability of the 6-vertex model. 2010. arXiv: 1010 . 5031 [math-ph]. url: https://arxiv.org/abs/1010.5031

    2010 arXiv

  9. [14]

    A Representation of Orientable Combinatorial 3-Manifolds

    W. B. R. Lickorish. “A Representation of Orientable Combinatorial 3-Manifolds”. In:Annals of Mathematics 76(1962),p.531. url: https://api.semanticscholar.org/CorpusID:122949359

    1962

  10. [15]

    Modifications and Cobounding Manifolds

    Andrew H. Wallace. “Modifications and Cobounding Manifolds”. In:Canadian Journal of Math- ematics 12 (1960), pp. 503–528.doi: 10.4153/CJM-1960-045-7

    1960 doi

  11. [16]

    A calculus for framed links inS3

    Robion Kirby. “A calculus for framed links inS3”. In:Inventiones mathematicae45.1 (Feb. 1978), pp. 35–56

    1978

  12. [17]

    Invariants of three manifolds via link polynomials and quantum groups

    N. Reshetikhin and V. G. Turaev. “Invariants of three manifolds via link polynomials and quantum groups”. In:Invent. Math.103 (1991), pp. 547–597.doi: 10.1007/BF01239527

    1991 doi

  13. [18]

    Higher dimensional algebra and topological quantum field theory

    J. C. Baez and J. Dolan. “Higher dimensional algebra and topological quantum field theory”. In: J. Math. Phys.36 (1995), pp. 6073–6105.doi: 10.1063/1.531236. arXiv: q-alg/9503002

    1995 arXiv

  14. [19]

    On the classification of topological field theories

    Jacob Lurie. “On the classification of topological field theories”. In: Current developments in mathematics 2008.1 (2008), pp. 129–280

    2008

  15. [20]

    Higher-dimensional algebra IV: 2-tangles

    John C. Baez and Laurel Langford. “Higher-dimensional algebra IV: 2-tangles”. In:Advances in Mathematics 180.2 (2003), pp. 705–764. issn: 0001-8708. doi: https : / / doi . org / 10 . 1016 / S0001- 8708(03)00018- 5. url: https://www.sciencedirect.com/science/article/pii/ S00018...

    2003

  16. [21]

    Polynomial invariants for smooth four-manifolds

    S.K. Donaldson. “Polynomial invariants for smooth four-manifolds”. In: Topology 29.3 (1990), pp. 257–315. issn: 0040-9383. doi: https://doi.org/10.1016/0040-9383(90)90001-Z . url: https://www.sciencedirect.com/science/article/pii/004093839090001Z

    1990

  17. [22]

    Lectures on Seiberg-Witten Invariants

    John Douglas Moore. Lectures on Seiberg-Witten Invariants. 2nd ed. Vol. Lecture Notes in Math- ematics. Springer-Verlag Berlin Heidelberg, Jan. 2001, pp. VIII, 121.doi: https://doi.org/10. 1007/978-3-540-40952-6

    2001

  18. [23]

    Invariantsof4–manifoldsfromKhovanov–Rozansky link homology

    ScottMorrison,KevinWalker,andPaulWedrich.“Invariantsof4–manifoldsfromKhovanov–Rozansky link homology”. In:Geometry & Topology(2019). url: https://api.semanticscholar.org/ CorpusID:198967556

    2019

  19. [24]

    The Kontsevich Integral

    S Chmutov and S Duzhin. “The Kontsevich Integral”. In: Acta Applicandae Mathematica66.2 (Apr. 2001), pp. 155–190

    2001

  20. [25]

    Skein lasagna modules and handle de- compositions

    Ciprian Manolescu, Kevin Walker, and Paul Wedrich. “Skein lasagna modules and handle de- compositions”. In:Advances in Mathematics(2022). url: https://api.semanticscholar.org/ CorpusID:249538143

    2022

  21. [26]

    Infinite-Dimensional Representations of 2-Groups

    John Baez et al. “Infinite-Dimensional Representations of 2-Groups”. In:Memoirs of the American Mathematical Society219.1032 (2012). doi: 10.1090/s0065-9266-2012-00652-6 . url: https: //doi.org/10.1090%2Fs0065-9266-2012-00652-6

    2012 doi

  22. [27]

    In: Forum Mathematicum23.3 (2011), pp

    Christoph Wockel. In: Forum Mathematicum23.3 (2011), pp. 565–610.doi: doi:10.1515/form. 2011.020. url: https://doi.org/10.1515/form.2011.020

    2011 doi

  23. [28]

    Four Equivalent Versions of Non-Abelian Gerbes

    Thomas Nickelsen Nikolaus and Konrad Waldorf. “Four Equivalent Versions of Non-Abelian Gerbes”. In: Pacific Journal of Mathematics 264 (2011), pp. 355–420. url: https : / / api . semanticscholar.org/CorpusID:55265704

    2011

  24. [29]

    Central extensions of smooth 2–groups and a finite-dimensional string 2–group

    Christopher-J Schommer-Pries. “Central extensions of smooth 2–groups and a finite-dimensional string 2–group”. In:Geometry & Topology15.2 (May 2011), pp. 609–676.issn: 1465-3060. doi: 10.2140/gt.2011.15.609. url: http://dx.doi.org/10.2140/gt.2011.15.609

    2011 doi

  25. [30]

    Adjusted parallel transport for higher gauge theories

    Hyungrok Kim and Christian Saemann. “Adjusted parallel transport for higher gauge theories”. In: J. Phys. A53.44 (2020), p. 445206.doi: 10.1088/1751- 8121/ab8ef2 . arXiv: 1911.06390 [hep-th]

    2020 arXiv

  26. [31]

    L8-Algebras of Classical Field Theories and the Batalin-Vilkovisky For- malism

    Branislav Jurčo et al. “L8-Algebras of Classical Field Theories and the Batalin-Vilkovisky For- malism”. In: Fortsch. Phys. 67.7 (2019), p. 1900025. doi: 10 . 1002 / prop . 201900025. arXiv: 1809.09899 [hep-th]

    2019 arXiv

  27. [32]

    4-D semistrict higher Chern-Simons theory I

    Emanuele Soncini and Roberto Zucchini. “4-D semistrict higher Chern-Simons theory I”. In:Jour- nal of High Energy Physics2014.10 (2014)

    2014

  28. [33]

    4-dChern-SimonsTheory:HigherGaugeSymmetryandHolographicAspects

    RobertoZucchini.“4-dChern-SimonsTheory:HigherGaugeSymmetryandHolographicAspects”. In: JHEP 06 (2021), p. 025.doi: 10.1007/JHEP06(2021)025. arXiv: 2101.10646 [hep-th]

    2021 arXiv

  29. [34]

    Higher gauge theory and integrability

    Hank Chen and Joaquin Liniado. “Higher gauge theory and integrability”. In:Phys. Rev. D110.8 (2024), p. 086017.doi: 10.1103/PhysRevD.110.086017. arXiv: 2405.18625 [hep-th]

    2024 arXiv

  30. [35]

    5d 2-Chern-Simons Theory and 3d Integrable Field Theories

    Alexander Schenkel and Beno^it Vicedo. “5d 2-Chern-Simons Theory and 3d Integrable Field Theories”. In:Commun. Math. Phys.405.12 (2024), p. 293.doi: 10.1007/s00220-024-05170-9. arXiv: 2405.08083 [hep-th]

    2024 arXiv

  31. [38]

    Drinfeld double symmetry of the 4d Kitaev model

    Hank Chen. “Drinfeld double symmetry of the 4d Kitaev model”. In:JHEP 09 (2023), p. 141. doi: 10.1007/JHEP09(2023)141. arXiv: 2305.04729 [cond-mat.str-el]. 47

    2023 arXiv

  32. [39]

    Four-dimensional topological field theory, Hopf categories, and the canonical bases

    Louis Crane and Igor Frenkel. “Four-dimensional topological field theory, Hopf categories, and the canonical bases”. In:J. Math. Phys.35 (1994), pp. 5136–5154.doi: 10.1063/1.530746 . arXiv: hep-th/9405183

    1994 arXiv

  33. [40]

    url: https://doi.org/10.1007/978-3-319-59939- 7_5

    doi: 10.1007/978-3-319-59939-7_5 . url: https://doi.org/10.1007/978-3-319-59939- 7_5

    doi

  34. [41]

    Hopf categories

    Eliezer Batista, Stefaan Caenepeel, and Joost Vercruysse. “Hopf categories”. In: Algebras and representation theory19 (2016), pp. 1173–1216

    2016

  35. [43]

    Monoidal Bicategories and Hopf Algebroids

    Brian Day and Ross Street. “Monoidal Bicategories and Hopf Algebroids”. In:Advances in Math- ematics 129.1 (1997), pp. 99–157.issn: 0001-8708. doi: https://doi.org/10.1006/aima.1997

    1997 doi

  36. [44]

    Gray categories with duals and their diagrams

    John W. Barrett, Catherine Meusburger, and Gregor Schaumann. “Gray categories with duals and their diagrams”. In:Advances in Mathematics450 (July 2024), p. 109740.issn: 0001-8708. doi: 10.1016/j.aim.2024.109740. url: http://dx.doi.org/10.1016/j.aim.2024.109740

    2024

  37. [45]

    3-dimensional defect TQFTs and their tricategories

    Nils Carqueville, Catherine Meusburger, and Gregor Schaumann. “3-dimensional defect TQFTs and their tricategories”. In:Adv. Math.364 (2020), p. 107024.doi: 10.1016/j.aim.2020.107024. arXiv: 1603.01171 [math.QA]

    2020

  38. [46]

    Categorified trace for module tensor categories over braided tensor categories

    David Penneys James Tener André Henriques. “Categorified trace for module tensor categories over braided tensor categories”. In:Doc. Math.21 (2016), pp. 1089–1149

    2016

  39. [47]

    Dagger n-categories

    Giovanni Ferrer et al. Dagger n-categories. 2024. arXiv: 2403.01651 [math.CT] . url: https: //arxiv.org/abs/2403.01651

    2024

  40. [48]

    Loop spaces, and coherence for monoidal and braided monoidal bicategories

    Nick Gurski. “Loop spaces, and coherence for monoidal and braided monoidal bicategories”. In: Advances in Mathematics226.5 (2011), pp. 4225–4265.issn: 0001-8708. doi: https://doi.org/ 10.1016/j.aim.2010.12.007. url: https://www.sciencedirect.com/science/article/pii/ S0001870810004408

    2011 doi

  41. [49]

    The center of monoidal 2-categories in 3+1D Dijkgraaf- Witten theory

    Liang Kong, Yin Tian, and Shan Zhou. “The center of monoidal 2-categories in 3+1D Dijkgraaf- Witten theory”. In:Advances in Mathematics360 (2020), p. 106928.issn: 0001-8708.doi: https: //doi.org/10.1016/j.aim.2019.106928 . url: https://www.sciencedirect.com/science/ article/pi...

    2020

  42. [51]

    M. Neuchl. Representation Theory of Hopf Categories. Verlag nicht ermittelbar, 1997.url: https: //books.google.ca/books?id=gpgLHAAACAAJ

    1997

  43. [52]

    Etingof et al

    P. Etingof et al. Tensor Categories. Mathematical Surveys and Monographs. American Math- ematical Society, 2016. isbn: 9781470434410. url: https : / / books . google . ca / books ? id = Z6XLDAAAQBAJ

    2016

  44. [53]

    Crossed Modules

    Friedrich Wagemann. Crossed Modules. De Gruyter, 2021. isbn: 9783110750959. doi: doi:10. 1515/9783110750959. url: https://doi.org/10.1515/9783110750959

    2021 doi

  45. [54]

    Strict quantum 2-groups

    Shahn Majid. “Strict quantum 2-groups”. In: (Aug. 2012). arXiv:1208.6265 [math.QA]

    2012 arXiv

  46. [55]

    A Combinatorial Description of Knotted Surfaces and Their Isotopies

    J.Scott Carter, Joachim H. Rieger, and Masahico Saito. “A Combinatorial Description of Knotted Surfaces and Their Isotopies”. In:Advances in Mathematics127.1 (1997), pp. 1–51.issn: 0001-

    1997

  47. [56]

    Higher-dimensional algebra. V: 2-Groups

    John C. Baez and Aaron D. Lauda. “Higher-dimensional algebra. V: 2-Groups.” eng. In:Theory and Applications of Categories [electronic only]12 (2004), pp. 423–491.url: http://eudml.org/ doc/124217. 48

    2004

  48. [57]

    Strict 2-Groups are Crossed Modules

    Sven-S. Porst. “Strict 2-Groups are Crossed Modules”. In:arXiv: Category Theory(2008)

    2008

  49. [58]

    Poisson 2-groups

    Zhuo Chen, Mathieu Stiénon, and Ping Xu. “Poisson 2-groups”. In:J. Diff. Geom.94.2 (2013), pp. 209–240. doi: 10.4310/jdg/1367438648. arXiv: 1202.0079 [math.DG]

    2013

  50. [59]

    Connections on non-abelian Gerbes and their Holonomy

    Urs Schreiber and Konrad Waldorf. “Connections on non-abelian Gerbes and their Holonomy”. In: Theory Appl. Categ.28.17 (2013), pp. 476–540. arXiv:0808.1923 [math.DG]

    2013 arXiv

  51. [60]

    HOPF ALGEBROIDS AND QUANTUM GROUPOIDS

    JIANG-HUA LU. “HOPF ALGEBROIDS AND QUANTUM GROUPOIDS”. In: International Journal of Mathematics 07.01 (1996), pp. 47–70. doi: 10 . 1142 / S0129167X96000050. eprint: https : / / doi . org / 10 . 1142 / S0129167X96000050. url: https : / / doi . org / 10 . 1142 / S0129167X96000050

    1996

  52. [61]

    Gauging the Gauge and Anomaly Resolution

    Hank Chen and Florian Girelli. “Gauging the Gauge and Anomaly Resolution”. In: (Nov. 2022). arXiv: 2211.08549 [hep-th]

    2022 arXiv

  53. [62]

    L 8-algebramodelsandhigherChern–Simonstheories

    PatriciaRitterandChristianSämann.“L 8-algebramodelsandhigherChern–Simonstheories”.In: Reviews in Mathematical Physics28.09(Oct.2016),p.1650021. doi: 10.1142/s0129055x16500215. url: https://doi.org/10.1142

    2016 doi

  54. [63]

    Hamiltonian analysis of the BFCG theory for the Poincaré 2-group

    A. Miković, M. A. Oliveira, and M. Vojinović. “Hamiltonian analysis of the BFCG theory for the Poincaré 2-group”. In: Class. Quant. Grav. 33.6 (2016), p. 065007. doi: 10 . 1088 / 0264 - 9381/33/6/065007. arXiv: 1508.05635 [gr-qc]

    2016 arXiv

  55. [64]

    poincaré 2-group and quantum gravity

    A. Mikovic and M. Vojinovic. “poincaré 2-group and quantum gravity”. In:Class. Quant. Grav. 29 (2012), p. 165003.doi: 10.1088/0264-9381/29/16/165003. arXiv: 1110.4694 [gr-qc]

    2012 arXiv

  56. [65]

    On Yetter’s invariant and an extension of the Dijkgraaf- Witten invariant to categorical groups

    Joao Faria Martins and Timothy Porter. “On Yetter’s invariant and an extension of the Dijkgraaf- Witten invariant to categorical groups”. In:Theor. Appl. Categor.18 (2007), pp. 118–150. arXiv: math/0608484

    2007 arXiv

  57. [66]

    Four-Dimensional BF theory with cosmological term as a topological quantum field theory

    John C. Baez. “Four-Dimensional BF theory with cosmological term as a topological quantum field theory”. In:Lett. Math. Phys.38 (1996), pp. 129–143. doi: 10.1007/BF00398315 . arXiv: q-alg/9507006

    1996 arXiv

  58. [67]

    John C. Baez. Higher Yang-Mills Theory. 2002. doi: 10.48550/ARXIV.HEP- TH/0206130 . url: https://arxiv.org/abs/hep-th/0206130

    2002 doi

  59. [68]

    On 2-Group Global Symmetries and their Anomalies

    Francesco Benini, Clay Córdova, and Po-Shen Hsin. “On 2-Group Global Symmetries and their Anomalies”. In: JHEP 03 (2019), p. 118.doi: 10.1007/JHEP03(2019)118 . arXiv: 1803.09336 [hep-th]

    2019 arXiv

  60. [69]

    Exploring 2-Group Global Symmetries

    Clay Córdova, Thomas T. Dumitrescu, and Kenneth Intriligator. “Exploring 2-Group Global Symmetries”. In:JHEP 02 (2019), p. 184.doi: 10.1007/JHEP02(2019)184. arXiv: 1802.04790 [hep-th]

    2019 arXiv

  61. [70]

    Polyhedron phase space using 2- groups: κ-Poincaré as a Poisson 2-group

    Florian Girelli, Matteo Laudonio, and Panagiotis Tsimiklis. “Polyhedron phase space using 2- groups: κ-Poincaré as a Poisson 2-group”. In: (May 2021). arXiv:2105.10616 [hep-th]

    2021 arXiv

  62. [71]

    Generalized higher connections and Yang-Mills

    Danhua Song et al. “Generalized higher connections and Yang-Mills”. In: (Dec. 2021). arXiv: 2112.13370 [hep-th]

    2021 arXiv

  63. [72]

    Generalized Global Symmetries

    Davide Gaiotto et al. “Generalized Global Symmetries”. In: JHEP 02 (2015), p. 172. doi: 10. 1007/JHEP02(2015)172. arXiv: 1412.5148 [hep-th]

    2015 arXiv

  64. [73]

    Classification of3`1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

    Tian Lan and Xiao-Gang Wen. “Classification of3`1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions”. In:Phys. Rev. X 9 (2 Apr. 2019), p. 021005. doi: 10.1103/PhysRevX.9.021005 . url: https://link.aps.org/doi/10.1103/PhysRevX.9. 021005

    2019 doi

  65. [74]

    Defects in the 3-dimensional toric code model form a braided fusion 2-category

    Liang Kong, Yin Tian, and Zhi-Hao Zhang. “Defects in the 3-dimensional toric code model form a braided fusion 2-category”. In:JHEP 12 (2020), p. 078.doi: 10.1007/JHEP12(2020)078. arXiv: 2009.06564 [cond-mat.str-el]

    2020 arXiv

  66. [75]

    Higher Chern-Simons based on (2-)crossed modules

    Danhua Song et al. “Higher Chern-Simons based on (2-)crossed modules”. In: Journal of High Energy Physics 2023.7 (July 2023). doi: 10.1007/jhep07(2023)207 . url: https://doi.org/ 10.1007%2Fjhep07%282023%29207

    2023 doi

  67. [76]

    Gauge Models of Topological Phases and Applications to Quantum Gravity

    Clement Delcamp. “Gauge Models of Topological Phases and Applications to Quantum Gravity”. PhD thesis. Waterloo U., 2018. 49

    2018

  68. [77]

    Topological phases from higher gauge symmetry in 3+1 dimensions

    A. Bullivant et al. “Topological phases from higher gauge symmetry in 3+1 dimensions”. In: Phys. Rev.B95.15 (2017), p. 155118.doi: 10.1103/PhysRevB.95.155118 . arXiv: 1606.06639 [cond-mat.str-el]

    2017 arXiv

  69. [78]

    Dynamics of a lattice 2-group gauge theory model

    A. Bochniak et al. “Dynamics of a lattice 2-group gauge theory model”. In:Journal of High Energy Physics 2021.9 (Sept. 2021).doi: 10.1007/jhep09(2021)068. url: https://doi.org/10.1007% 2Fjhep09%282021%29068

    2021 doi

  70. [79]

    Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d

    Ryan Thorngren. “Framed Wilson operators, fermionic strings, and gravitational anomaly in 4d”. In: Journal of High Energy Physics 2015.152 (2010). doi: 10 . 1007 / JHEP02(2015 ) 152. url: https://doi.org/10.1007/JHEP02(2015)152

    2010 doi

  71. [80]

    Twisted gauge theories in three-dimensional Walker-Wang models

    Zitao Wang and Xie Chen. “Twisted gauge theories in three-dimensional Walker-Wang models”. In: Phys. Rev.B95.11 (2017), p. 115142.doi: 10.1103/PhysRevB.95.115142. arXiv:1611.09334 [cond-mat.str-el]

    2017 arXiv

  72. [81]

    Differential twisted String and Fivebrane struc- tures

    Hisham Sati, Urs Schreiber, and Jim Stasheff. “Differential twisted String and Fivebrane struc- tures”. In:Commun. Math. Phys.315 (2012), pp. 169–213.doi: 10.1007/s00220-012-1510-3 . arXiv: 0910.4001 [math.AT]

    2012 arXiv

  73. [82]

    Differential cohomology in a cohesive infinity-topos

    Urs Schreiber. “Differential cohomology in a cohesive infinity-topos”. In: (Oct. 2013). arXiv:1310. 7930 [math-ph]

    2013

  74. [83]

    Higher-Dimensional Algebra VI: Lie 2-Algebras

    John C. Baez and Alissa S. Crans. “Higher-Dimensional Algebra VI: Lie 2-Algebras”. In:Theor. Appl. Categor.12 (2004), pp. 492–528. arXiv:math/0307263

    2004 arXiv

  75. [84]

    Nonabelian Multiplicative Integration on Surfaces.WORLDSCIENTIFIC,2015

    AmnonYekutieli. Nonabelian Multiplicative Integration on Surfaces.WORLDSCIENTIFIC,2015. doi: 10.1142/9537. eprint: https://www.worldscientific.com/doi/pdf/10.1142/9537. url: https://www.worldscientific.com/doi/abs/10.1142/9537

    2015 doi

  76. [85]

    Higher-form Gauge Symmetries in Mul- tipole Topological Phases

    Oleg Dubinkin, Alex Rasmussen, and Taylor L. Hughes. “Higher-form Gauge Symmetries in Mul- tipole Topological Phases”. In:Annals Phys.422 (2020), p. 168297.doi: 10.1016/j.aop.2020. 168297. arXiv: 2007.05539 [cond-mat.str-el]

    2020 arXiv

  77. [86]

    Combinatorial quantization of the Hamiltonian Chern-Simons theory

    Anton Yu. Alekseev, Harald Grosse, and Volker Schomerus. “Combinatorial quantization of the Hamiltonian Chern-Simons theory”. In: Commun. Math. Phys. 172 (1995), pp. 317–358. doi: 10.1007/BF02099431. arXiv: hep-th/9403066 [hep-th]

    1995 arXiv

  78. [87]

    Combinatorial quantization of the Hamiltonian Chern-Simons theory. 2

    Anton Yu. Alekseev, Harald Grosse, and Volker Schomerus. “Combinatorial quantization of the Hamiltonian Chern-Simons theory. 2.” In:Commun. Math. Phys.174 (1995), pp. 561–604.doi: 10.1007/BF02101528. arXiv: hep-th/9408097 [hep-th]

    1995 arXiv

  79. [88]

    Dynamical generalization of Yetter’s model based on a crossed module of discrete groups

    Arkadiusz Bochniak, Leszek Hadasz, and B. Lażej Ruba. “Dynamical generalization of Yetter’s model based on a crossed module of discrete groups”. In:JHEP 03 (2021), p. 282.doi: 10.1007/ JHEP03(2021)282. arXiv: 2010.00888 [math-ph]

    2021 arXiv

  80. [89]

    Higher Symmetry and Gapped Phases of Gauge Theories

    Anton Kapustin and Ryan Thorngren. “Higher Symmetry and Gapped Phases of Gauge Theories”. In: Algebra, Geometry, and Physics in the 21st Century: Kontsevich Festschrift. Ed. by Denis Auroux et al. Cham: Springer International Publishing, 2017, pp. 177–202.isbn: 978-3-319-59939-

    2017

  81. [90]

    Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons inD4

    Hank Chen. “Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons inD4”. In: (June 2025). arXiv:2506.05785 [math-ph]

    2025

  82. [91]

    Measurable Categories

    David N. Yetter. “Measurable Categories”. In:Applied Categorical Structures13 (2003), pp. 469–

    2003

  83. [92]

    Monoidal categories graded by crossed modules and 3- dimensional HQFTs

    Kürsat Sözer and Alexis Virelizier. “Monoidal categories graded by crossed modules and 3- dimensional HQFTs”. In:Advances in Mathematics428 (2023), p. 109155.issn: 0001-8708. doi: https://doi.org/10.1016/j.aim.2023.109155 . url: https://www.sciencedirect.com/ science/article/p...

    2023

  84. [93]

    Lie 2-Bialgebras

    Chengming Bai, Yunhe Sheng, and Chenchang Zhu. “Lie 2-Bialgebras”. In:Communications in Mathematical Physics320.1 (Apr. 2013), pp. 149–172.doi: 10.1007/s00220-013-1712-3 . url: https://doi.org/10.1007%2Fs00220-013-1712-3. 50

    2013 doi

  85. [94]

    Categorified Drinfel’d double andBF theory: 2-groups in 4D

    Hank Chen and Florian Girelli. “Categorified Drinfel’d double andBF theory: 2-groups in 4D”. In: Phys. Rev. D 106 (10 Nov. 2022), p. 105017.doi: 10.1103/PhysRevD.106.105017 . url: https://link.aps.org/doi/10.1103/PhysRevD.106.105017

    2022 doi

  86. [96]

    Measurable Categories and 2-Groups

    Louis Crane and David N Yetter. “Measurable Categories and 2-Groups”. In:Applied Categorical Structures 13.5 (Dec. 2005), pp. 501–516

    2005

  87. [97]

    Representation theory for categor- ical symmetries

    Thomas Bartsch, Mathew Bullimore, and Andrea Grigoletto. “Representation theory for categor- ical symmetries”. In: (May 2023). arXiv:2305.17165 [hep-th]

    2023 arXiv

  88. [98]

    The 2-character theory for finite 2-groups

    Mo Huang, Hao Xu, and Zhi-Hao Zhang. “The 2-character theory for finite 2-groups”. In:arXiv e-prints, arXiv:2404.01162 (Apr. 2024), arXiv:2404.01162.doi: 10.48550/arXiv.2404.01162 . arXiv: 2404.01162 [math.RT]

    2024 doi arXiv

  89. [99]

    Highercategoricalsymmetriesandgaugingintwo-dimensional spin systems

    ClementDelcampandApoorvTiwari.“Highercategoricalsymmetriesandgaugingintwo-dimensional spin systems”. In:SciPost Phys. 16.4 (2024), p. 110. doi: 10.21468/SciPostPhys.16.4.110 . arXiv: 2301.01259 [hep-th]

    2024 arXiv

  90. [100]

    Higher-Dimensional Algebra II. 2-Hilbert Spaces

    John C. Baez. “Higher-Dimensional Algebra II. 2-Hilbert Spaces”. In:Advances in Mathematics 127 (1996), pp. 125–189.url: https://api.semanticscholar.org/CorpusID:2792589

    1996

  91. [101]

    Representation and character theory in 2-categories

    Nora Ganter and Mikhail Kapranov. “Representation and character theory in 2-categories”. In: arXiv Mathematics e-prints, math/0602510 (Feb. 2006), math/0602510.doi: 10.48550/arXiv. math/0602510. arXiv: math/0602510 [math.KT]

    2006 doi arXiv

  92. [102]

    2-categoriesandZamolodchikovtetrahedraequations

    M.M.KapranovandV.A.Voevodsky.“2-categoriesandZamolodchikovtetrahedraequations”.In: Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991). Vol. 56. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1994, 177–259

    1991

  93. [103]

    Non-invertible symmetries and higher representation theory I

    Thomas Bartsch et al. “Non-invertible symmetries and higher representation theory I”. In:SciPost Phys. 17.1 (2024), p. 015.doi: 10.21468/SciPostPhys.17.1.015. arXiv:2208.05993 [hep-th]

    2024 arXiv

  94. [104]

    The Eckmann–Hilton argument and higher operads

    M.A. Batanin. “The Eckmann–Hilton argument and higher operads”. In:Advances in Mathematics 217.1 (2008), pp. 334–385.issn: 0001-8708. doi: https://doi.org/10.1016/j.aim.2007.06

    2008 doi

  95. [105]

    Manifestly unitary higher Hilbert spaces

    Quan Chen et al. “Manifestly unitary higher Hilbert spaces”. In: 2024. url: https : / / api . semanticscholar.org/CorpusID:273186640

    2024

  96. [106]

    Operator Algebras in Rigid C*-Tensor Categories

    Corey Jones and David Penneys. “Operator Algebras in Rigid C*-Tensor Categories”. In:Com- munications in Mathematical Physics355.3 (Nov. 2017), pp. 1121–1188.doi: 10.1007/s00220- 017-2964-0. arXiv: 1611.04620 [math.OA]

    2017 arXiv

  97. [107]

    Dagger categories via anti-involutions and positivity

    Luuk Stehouwer and Jan Steinebrunner. “Dagger categories via anti-involutions and positivity”. In: arXiv preprint arXiv:2304.02928(2023)

    2023 arXiv

  98. [108]

    Spherical Categories

    John W Barrett and Bruce W Westbury. “Spherical Categories”. In:Advances in Mathematics 143.2 (1999), pp. 357–375.issn: 0001-8708. doi: https://doi.org/10.1006/aima.1998.1800 . url: https://www.sciencedirect.com/science/article/pii/S000187089891800X

    1999

  99. [109]

    Catégories tannakiennes

    P. Deligne. “Catégories tannakiennes”. In:The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck. Ed. by Pierre Cartier et al. Boston, MA: Birkhäuser Boston, 2007, pp. 111–195.isbn: 978-0-8176-4575-5. doi: 10.1007...

    2007 doi

  100. [110]

    M.N. Gurski. An Algebraic Theory of Tricategories. University of Chicago, Department of Math- ematics, 2006. url: https://books.google.ca/books?id=HTwCOAAACAAJ

    2006

  101. [111]

    (Co)end Calculus

    Fosco Loregian. (Co)end Calculus. London Mathematical Society Lecture Note Series. Cambridge University Press, 2021. 51

    2021

  102. [112]

    url: https://www.sciencedirect.com/science/article/pii/S0001870807002113

  103. [113]

    Fusion categories via string diagrams

    Bruce Bartlett. “Fusion categories via string diagrams”. In: Communications in Contemporary Mathematics 18.05 (2016), p. 1550080.doi: 10.1142/S0219199715500807. eprint:https://doi. org/10.1142/S0219199715500807. url: https://doi.org/10.1142/S0219199715500807

    2016 doi

  104. [114]

    OntheantipodeofaquasitriangularHopfalgebra

    DavidERadford.“OntheantipodeofaquasitriangularHopfalgebra”.In: Journal of Algebra151.1 (1992), pp. 1–11. issn: 0021-8693. doi: https://doi.org/10.1016/0021- 8693(92)90128- 9 . url: https://www.sciencedirect.com/science/article/pii/0021869392901289

    1992

  105. [115]

    The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

    David E. Radford. “The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite”. In: American Journal of Mathematics98.2 (1976), pp. 333–355.issn: 00029327, 10806377. url: http://www.jstor.org/stable/2373888 (visited on 12/13/2024)

    1976

  106. [116]

    A note on Radford’s S4 formula

    Lydia Delvaux, Alfons Van Daele, and Shuanhong Wang. “A note on Radford’s S4 formula”. In: arXiv: Rings and Algebras(2006). url: https://api.semanticscholar.org/CorpusID: 1770960

    2006

  107. [117]

    Minimal nondegenerate extensions

    Theo Johnson-Freyd and David Reutter. “Minimal nondegenerate extensions”. In:Journal of the American Mathematical Society(July 2023). issn: 1088-6834. doi: 10.1090/jams/1023 . url: http://dx.doi.org/10.1090/jams/1023

    2023 doi

  108. [118]

    Finite groups, spherical 2-categories, and 4-manifold invariants

    Marco Mackaay. “Finite groups, spherical 2-categories, and 4-manifold invariants”. In:Adv. Math. 153(2) (2000), 353-390 (2000). eprint: math/9903003. url: https://arxiv.org/pdf/math/ 9903003.pdf

    2000 arXiv

  109. [119]

    An analogue of Radford’s S4 formula for finite tensor categories

    Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. “An analogue of Radford’s S4 formula for finite tensor categories”. In:International Mathematics Research Notices2004.54 (2004), pp. 2915–2933. doi: 10.1155/S1073792804141445

    2004 doi

  110. [120]

    Tannakian Categories

    P. Deligne and J. S. Milne. “Tannakian Categories”. In: Hodge Cycles, Motives, and Shimura Varieties. Berlin, Heidelberg: Springer Berlin Heidelberg, 1982, pp. 101–228. isbn: 978-3-540- 38955-2. doi: 10.1007/978-3-540-38955-2_4 . url: https://doi.org/10.1007/978-3-540- 38955-2_4

    1982 doi

  111. [121]

    Domain Walls Between 3d Phases of Reshetikhin–Turaev TQFTs

    Vincent Koppen et al. “Domain Walls Between 3d Phases of Reshetikhin–Turaev TQFTs”. In: Commun. Math. Phys.396.3 (2022), pp. 1187–1220.doi: 10.1007/s00220-022-04489-5. arXiv: 2105.04613 [hep-th]

    2022 arXiv

  112. [122]

    Orbifold completion of 3-categories

    Nils Carqueville and Lukas Müller. “Orbifold completion of 3-categories”. In: (July 2023). arXiv: 2307.06485 [math.QA]

    2023

  113. [123]

    Categories of quantum liquids I

    Liang Kong and Hao Zheng. “Categories of quantum liquids I”. In:JHEP 08 (2022), p. 070.doi: 10.1007/JHEP08(2022)070. arXiv: 2011.02859 [hep-th]

    2022 arXiv

  114. [124]

    Categories of Quantum Liquids II

    Liang Kong and Hao Zheng. “Categories of Quantum Liquids II”. In:Communications in Mathe- matical Physics405.9 (Aug. 2024), p. 203

    2024

  115. [125]

    HigherDimensionalAlgebra:I.BraidedMonoidal2-Categories

    JohnCBaezandMartinNeuchl.“HigherDimensionalAlgebra:I.BraidedMonoidal2-Categories”. In: Advances in Mathematics121.2 (1996), pp. 196–244.issn: 0001-8708.doi: https://doi.org/ 10.1006/aima.1996.0052 . url: https://www.sciencedirect.com/science/article/pii/ S0001870896900523

    1996

  116. [126]

    2-Tangles

    John C Baez and Laurel Langford. “2-Tangles”. In:Letters in Mathematical Physics43 (1998), pp. 187–197

    1998

  117. [127]

    Algebraic higher symmetry and categorical symmetry: A holographic and en- tanglement view of symmetry

    Liang Kong et al. “Algebraic higher symmetry and categorical symmetry: A holographic and en- tanglement view of symmetry”. In:Phys. Rev. Research2 (4 Oct. 2020), p. 043086.doi: 10.1103/ PhysRevResearch.2.043086. url: https://link.aps.org/doi/10.1103/PhysRevResearch. 2.043086

    2020 doi

  118. [128]

    Orbifolds of Reshetikhin-Turaev TQFTs

    Nils Carqueville, Ingo Runkel, and Gregor Schaumann. “Orbifolds of Reshetikhin-Turaev TQFTs”. In: Theor. Appl. Categor.35 (2020), pp. 513–561. arXiv:1809.01483 [math.QA]

    2020 arXiv

  119. [129]

    A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity

    Alberto S. Cattaneo and Alejandro Perez. “A note on the Poisson bracket of 2d smeared fluxes in loop quantum gravity”. In:Class. Quant. Grav.34.10 (2017), p. 107001.doi: 10.1088/1361- 6382/aa69b4. arXiv: 1611.08394 [gr-qc]. 52

    2017 arXiv

  120. [130]

    Gauge theories, vertex models, and quantum groups

    Edward Witten. “Gauge theories, vertex models, and quantum groups”. In: Nuclear Physics B 330.2 (1990), pp. 285–346. issn: 0550-3213. doi: https://doi.org/10.1016/0550- 3213(90) 90115-T. url: https://www.sciencedirect.com/science/article/pii/055032139090115T. 53

    1990

  121. [133]

    Condensations in higher categories

    Davide Gaiotto and Theo Johnson-Freyd. “Condensations in higher categories”. In: (May 2019). arXiv: 1905.09566 [math.CT]

    2019 arXiv

  122. [134]

    Models for Gapped Boundaries and Domain Walls

    Alexei Kitaev and Liang Kong. “Models for Gapped Boundaries and Domain Walls”. In:Commu- nications in Mathematical Physics313.2 (June 2012), pp. 351–373.doi: 10.1007/s00220-012- 1500-5. url: https://doi.org/10.1007%2Fs00220-012-1500-5

    2012 doi

  123. [136]

    Non-semisimple WRT at the boundary of Crane-Yetter

    Benjamin Haïoun. Non-semisimple WRT at the boundary of Crane-Yetter. 2025. arXiv: 2503. 20905 [math.QA]. url: https://arxiv.org/abs/2503.20905

    2025 arXiv

  124. [500]

    url: https://api.semanticscholar.org/CorpusID:15178814

  125. [1649]

    url: https://www.sciencedirect.com/science/article/pii/S0001870897916492

  126. [2012]

    arXiv: 1206.2308 [math.QA]

    arXiv

  127. [8708]

    url: https://www.sciencedirect

    doi: https://doi.org/10.1006/aima.1997.1618 . url: https://www.sciencedirect. com/science/article/pii/S0001870897916182

    1997

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.