A Larson-Sweedler Theorem for Hopf V-Categories

Christina Vasilakopoulou; Joost Vercruysse; Mitchell Buckley; Timmy Fieremans

REVIEW 4 major objections 5 minor 13 references

A Larson-Sweedler Theorem for Hopf V-Categories

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A many-object generalization of the Larson-Sweedler theorem is proved for Hopf V-categories.

desk verdict A substantial many-object generalization of Larson-Sweedler that mostly delivers, with a real but manageable dependency on prior work and a small gap in one corollary. read the letter →

arxiv 1908.02049 v3 pith:5LXLYVC6 submitted 2019-08-06 math.CT math.RA

Mitchell Buckley , Timmy Fieremans , Christina Vasilakopoulou , Joost Vercruysse This is my paper · ORCID

classification math.CTmath.RA MSC 18D2016T0518M05

keywords Larson-SweedlertheoremHopfV-categoriesFrobeniusintegraltheorymonoidsmany-objectalgebrasnon-singularintegralsTuraevgroup

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

This paper proves a many-object version of the classical theorem that a finite-dimensional bialgebra is a Hopf algebra exactly when it admits a non-singular integral. The setting is a Hopf V-category, a category enriched over comonoids in a braided monoidal category V, with a Hopf algebra as the one-object case; the paper develops integral families and integral spaces for these categories and shows that when every hom-object is dualizable (local rigidity), 'Hopf plus a non-singular right integral' is equivalent to 'Hopf plus Frobenius' and to 'integral spaces isomorphic to the monoidal unit.' If correct, one uniform theorem covers classical Hopf algebras, Hopf monoids in braided categories, Turaev Hopf group algebras, and packed forms producing weak and multiplier Hopf algebras.

What carries the argument

The load-bearing objects are Hopf V-categories: categories enriched over the monoidal category of comonoids in a braided monoidal category V, so each hom-object $A_{x,y}$ is a comonoid with local comultiplication and counit, and an antipode $s_{xy}: A_{x,y} \to A_{y,x}$. Integral families are morphisms from the unit graph to the regular module, and integral spaces are defined as limits, not equalizers; non-singularity means the induced maps $p_{xx}$ and $q_{xx}$ on internal hom objects are split epimorphisms. The proof chain uses the fundamental theorem of Hopf modules (imported from earlier work), a dual fundamental theorem for Hopf opmodules proved here, and new characterizations of Frobenius V-categories by Casimir families, module isomorphisms $A \cong A^{*,\mathrm{op}}$, non-degenerate trace pairings, and Frobenius adjunctions.

What would settle it

Find a locally rigid semi-Hopf V-category over $\mathbf{Vect}_k$ that has non-singular left and right integral families but whose antipode is not invertible, or whose left integral space is not isomorphic to $k$; either example would directly contradict Theorem 4.16(iv) and Theorem 4.17. Equivalently, exhibit a Hopf V-category whose regular Hopf module violates the coinvariant reconstruction of the fundamental theorem.

Watch

Extended reading notes

Core claim

The central result, Theorem 4.17, states that for a locally rigid semi-Hopf V-category A the following are equivalent: (i) A is Hopf and has a non-singular right integral family; (ii) A has both a non-singular right and a non-singular left integral family; (iii) A is Hopf and Frobenius; (iv) A is Hopf and the left integral spaces $\int^\ell A_x$ are isomorphic to the monoidal unit $I$ for all $x$; plus the dual statements for the dual semi-Hopf V-opcategory and the left-right interchanged versions. Over k-modules where projective modules are free, the list is further equivalent to A being simply Hopf (Corollary 4.19), which is exactly the classical Larson-Sweedler statement in the one-object case.

Load-bearing premise

The load-bearing premise is an imported theorem, used with only a proof sketch: for every Hopf module over a Hopf V-category, the coinvariants tensored back with the category reconstruct the original module ($M^{coA} \otimes A \cong M$). If that reconstruction fails, the proofs of antipode invertibility and of the Frobenius equivalence collapse.

Editorial extensions

If this is right

In the one-object case, the theorem yields a Larson-Sweedler statement for Hopf monoids in any braided monoidal category, including monoidal Hom-Hopf algebras, graded Hopf algebras, and Yetter-Drinfel'd Hopf algebras.
For a k-linear Hopf category with finitely many objects and finite-dimensional hom-objects, the packed form is a Frobenius weak Hopf algebra; with infinitely many objects it gives a Frobenius weak multiplier Hopf algebra.
Finite-dimensional Turaev Hopf G-algebras become Frobenius categories, and the paper spells out explicit cocomposition and trace maps for the associated Hopf category and for groupoid algebras.
Over rings where every finitely generated projective module is free, the conditions collapse: A is Hopf if and only if it has a right non-singular left integral family and the non-zero hom-objects have equal dimensions (Corollary 4.19).
Every Frobenius Hopf V-category is locally Frobenius: each hom-object $H_{x,y}$ is itself a Frobenius algebra in V, so the local comonoid and local monoid structures coexist on the same objects.

Reading between the lines

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

The theorem suggests that 'many-object quantum groups' could be defined by integral existence rather than by antipodes, in the same way the classical theorem motivated locally compact quantum groups; this is a natural programmatic next step, not stated in the paper.
Because non-singularity is tested only on diagonal hom-objects $p_{xx}$ and $q_{xx}$ yet Theorem 4.16 shows all $p_{xy}$ and $q_{xy}$ become isomorphisms, there is a local-to-global principle worth testing: checking endo-hom components alone may certify integral non-singularity everywhere.
Replacing the object-indexing set $X \times X$ by an arbitrary groupoid could unify Hopf G-algebras with Hopf categories and yield one common Larson-Sweedler theorem; the paper names this as a direction for future work.
The identification of locally rigid Calabi-Yau categories with symmetric Frobenius categories (Corollary 3.22) suggests integral data could serve as a categorical trace in TQFT constructions, beyond anything the paper proves.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a genuine many-object Larson-Sweedler theorem, and the proof structure looks right. I would send it to a serious referee. The biggest caveat is exactly the one the stress-test flags: the main chain uses the fundamental theorem of Hopf modules (Thm 2.24) as a black box from [BCV16], with only a sketch and a pointer to Böhm-Lack. That is a published result and the pointer is reassuring, but the equivalence in Theorem 4.17 cannot be checked without trusting it, so a referee should verify the hypotheses in [BCV16]. The paper would be stronger with a full proof or at least a precise statement of the hypotheses.

What is new: the multi-object theorem itself, the treatment of integral spaces as limits rather than equalizers, and the Frobenius characterizations in Section 3 are real contributions. The one-object folklore case is honestly labelled. The applications to Turaev and weak multiplier Hopf algebras look illustrative rather than forced. The self-citations are appropriate; the paper builds directly on [BCV16] and [BFVV17].

Where the soft spots are: Corollary 4.19 has a cancellation step (A*xx ≅ ∫ℓ A,x ⊗ Axx implies ∫ℓ A,x has rank one). It is probably fine, but the authors should say that ∫ℓ A,x is finitely generated projective; otherwise the rank argument is implicit. Minor. Also, non-singularity is defined through split epimorphisms rather than isomorphisms; this is fine for fields but deserves a remark in the concrete cases. The proof of Proposition 4.9 is more of a diagram sketch than a full verification, but I do not see a substantive gap.

The stress-test note overstates the 'outsourced' problem: importing a published theorem from a previous paper is standard practice. But it is fair to say the main result inherits any hidden hypotheses in that theorem. The paper explicitly omits the converse version of the opmodule fundamental theorem, saying it is not needed; that is accurate as far as I can tell.

Who this is for: Hopf algebra people and enriched category theorists. The paper unifies a lot of specialized results and gives a clean framework. It deserves referee time.

Referee Report

4 major / 5 minor

Summary. The paper develops a many-object generalization of the classical Larson-Sweedler theorem. Working with Hopf V-categories, i.e. categories enriched over comonoids in a braided monoidal category V, the authors introduce integral families and integral spaces, prove several characterizations of Frobenius V-categories, and establish Theorem 4.17: for a locally rigid semi-Hopf V-category, being Hopf with a non-singular right integral family is equivalent to having both non-singular left and right integral families, to being Hopf and Frobenius, to being Hopf with trivial left integral spaces, and to the dual statements for the dual opcategory. The paper also derives a version over rings with all projective modules free (Corollary 4.19) and discusses applications to Hopf algebras in braided categories, groupoid algebras, Turaev's Hopf group-coalgebras, and weak (multiplier) Hopf algebras. The proof is structured and heavily diagrammatic, with an integral theory modeled on the one-object case but adapted to the many-object setting.

Significance. If the result is correct, it is a substantial generalisation: it unifies the classical Larson-Sweedler theorem, its braided-monoidal one-object version, and several related results for weak and multiplier Hopf algebras, and it gives a categorical explanation of the integral/Frobenius correspondence. The paper also provides genuinely useful characterizations of Frobenius V-categories in terms of Casimir families, dual modules, trace maps, and Frobenius functors, and it contains explicit formulas for the antipode and Frobenius isomorphisms constructed from integrals. The examples in Section 5, especially those involving Turaev Hopf group-coalgebras and groupoid algebras, add concrete value. The main limitation is that the central theorem inherits a substantial dependency on the fundamental theorem of Hopf modules imported from earlier work with only a proof sketch; the reader cannot fully verify the load-bearing input from the manuscript alone.

major comments (4)

[§2.2 (Theorem 2.24)] Theorem 2.24, the fundamental theorem of Hopf modules for Hopf V-categories, is stated with only a proof sketch: the unit and counit are named, but the verification that they form an adjoint equivalence, and the proof of the 'if and only if' claim, are not given. This theorem is used essentially in Lemma 2.25 (invertibility of the antipode), in Theorem 4.13, and in Corollary 4.19. Since the statement as written assumes only that V has equalizers, whereas the paper elsewhere needs limits or completeness, the precise hypotheses of the imported theorem should be stated and either a complete proof supplied or a precise pointer to the proof in [BCV16] must be given. This is a load-bearing dependence, not a stylistic matter.
[§4.3 (Theorem 4.13, (ii)⇒(iii))] In the proof of (ii)⇒(iii), the argument passes from a right H*_{y,x}-module isomorphism H*_{y,x} ≅ H_{y,x} to the conclusion that H*_{y,x} is a Frobenius monoid via the one-object case of Proposition 3.12. This step also needs the identification of the dual of H*_{y,x} with H_{y,x}, which follows from local rigidity but is not explicitly stated or proved. Please add the missing identification so that the application of Proposition 3.12 is transparent.
[§4.4 (Corollary 4.19, (viii)⇒(iv))] The proof of (viii)⇒(iv) concludes from A*_{x,x} ≅ ∫ℓ A,x ⊗ A_{x,x} and the equality of dimensions that ∫ℓ A,x is free of rank one. This cancellation step is valid only after one observes that ∫ℓ A,x is a direct summand of the free module A*_{x,x} (or otherwise justifies projectivity), so that the ring hypothesis applies. The argument is easily repairable, but as written it is a compressed step in a key corollary and should be spelled out.
[§4 (Theorem 4.17 statement)] Theorem 4.17 is stated for a locally rigid semi-Hopf V-category, but the notions of non-singular integral (Definition 4.4) and integral space (Definition 4.7) require V to be braided monoidal closed and to have limits. These standing hypotheses should be incorporated into the theorem statement, or at least explicitly recalled there, so that the statement is self-contained.

minor comments (5)

[§4.3 (Theorem 4.16(iv), displayed computation)] The proof of Theorem 4.16(iv) contains apparent notation errors: in the displayed verification, composites such as qxx◦qxx and pxx◦pxx are ill-typed; the intended maps appear to be the right inverses qx and px. Please correct these composites.
[§4.2 and §4.3 (notation H* vs. H*,op)] The notation H* is used in Theorem 4.13(iv), Proposition 4.10, and elsewhere without consistently indicating whether it means the dual V-graph, the dual V-opcategory H*,op, or the associated semi-Hopf V-opcategory. Please fix the notation, for example by writing H*,op whenever an opcategory is meant.
[§2.3 and §4.2 (Proposition 4.9)] Proposition 4.9 asserts a natural isomorphism between the diagrams (41) and (42) but does not fully verify that the proposed morphisms commute with all legs of the limit diagrams. Adding the explicit compatibility check would make the isomorphism of integral spaces rigorous and easier to follow.
[§4.4 (Lemma 4.18 and Corollary 4.19)] The phrase 'same dimension' in Lemma 4.18 and Corollary 4.19 should be defined precisely for modules over the base ring k; the paper's convention that finite dimensional means finitely generated projective of finite rank is recalled, but the dimension equalities are used in contexts where nonzero hom-objects are involved, and a short clarification would prevent ambiguity.
[Throughout] Several displayed commutative diagrams are visually dense and some labels (e.g. in the proof of Theorem 4.13 and Remark 4.14) are nearly illegible in the text version. Reproducing the diagrams with clearer placement of tensor factors and labels would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 4.17 is a genuine extension; the load-bearing import of Theorem 2.24 from [BCV16] is independent support, not a circular step.

full rationale

The paper's main theorem (4.17) is not derived from its conclusion. Integral families (Def. 4.1) and non-singularity (Def. 4.4) are defined purely from the semi-Hopf data (m, j, δ, ε) without reference to an antipode or a Frobenius structure; antipodes are then constructed from integral data in Thm 4.16, and Frobenius equivalence is established through Casimir families, module isomorphisms, and the fundamental theorems. The closest thing to a circular dependency is the use of Thm 2.24, the fundamental theorem of Hopf modules for Hopf V-categories, imported from [BCV16] with only a proof sketch, and the paper also explicitly omits the converse of the opmodule version in Section 2.3. This is a genuine load-bearing external result with overlapping authorship (J. Vercruysse is a coauthor of [BCV16]), and if Thm 2.24 had unstated hypotheses the chain of implications in Lemma 2.25 and Thm 4.13 could collapse. That is a correctness risk, not circularity: [BCV16] is a published source with complete proofs, the theorem does not assume the Larson-Sweedler result, and the paper notes an independent route through [BL16]. No fitted input is renamed as a prediction, no uniqueness principle is imported to forbid alternatives, and no definition is circular. Hence score 0.

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

The paper introduces no new entities and has no free parameters. It relies on the prior notions of Hopf and Frobenius V-categories and on the fundamental theorem of Hopf modules from the authors' earlier works, which are published or forthcoming.

assumptions (4)

domain assumption V is a braided monoidal category that is monoidal closed and complete, or at least has the limits needed for integral spaces.
Integral spaces in Definition 4.7 are limits of diagrams in V; the main theorem assumes these exist and are well-behaved.
domain assumption A is locally rigid, meaning every hom-object Ax,y has a dual in V.
Used to define opposite dual graphs, to prove invertibility of the antipode in Lemma 2.25, and to establish Frobenius equivalence in Theorem 4.13.
standard math The fundamental theorem of Hopf modules for Hopf V-categories (Theorem 2.24), from [BCV16].
The paper's proofs of Lemma 2.25 and Theorem 4.13 depend on this theorem; only a proof sketch is given in the text.
domain assumption For Corollary 4.19, the base ring k has the property that all finitely generated projective modules are free, and the implicit cancellation of finite free modules in tensor products is valid.
Used to reduce Theorem 4.17 to the classical Larson-Sweedler theorem; the cancellation step from A*x,x ≅ ∫ℓ A,x ⊗ Ax,x to dim ∫ℓ A,x = 1 is not justified in the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Larson-Sweedler Theorem for Hopf V-Categories." pith.science (2026). https://pith.science/paper/5LXLYVC6

@misc{pith2026190802049,
  author       = {Pith},
  title        = {Pith review of: A Larson-Sweedler Theorem for Hopf V-Categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LXLYVC6}},
  note         = {Machine review of arXiv:1908.02049}
}

read the original abstract

The aim of this paper is to extend the classical Larson-Sweedler theorem, namely that a k-bialgebra has a non-singular integral (and in particular is Frobenius) if and only if it is a finite dimensional Hopf algebra, to the `many-object' setting of Hopf categories. To this end, we provide new characterizations of Frobenius V-categories and we develop the integral theory for Hopf V-categories. Our results apply to Hopf algebras in any braided monoidal category as a special case, and also relate to Turaev's Hopf group algebras and particular cases of weak and multiplier Hopf algebras.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Oplax Hopf Algebras

[BCV16] E. Batista, S. Caenepeel, and J. Vercruysse. “Hopf Cate gories”. In: Algebras and Representation Theory 19.5 (2016), pp. 1173–1216. [BFVV17] M. Buckley, T. Fieremans, C. Vasilakopoulou, and J. Vercr uysse. “Oplax Hopf Algebras”. arXiv:1710.01465 [math.CT], to appear in Higher Struc- tures

work page Pith review arXiv 2016
[10]

Hopf Galois ex tensions, smash products, and Morita equivalence

Topology and physics. 1994, pp. 5136–5154. [CFM90] M. Cohen, D. Fischman, and S. Montgomery. “Hopf Galois ex tensions, smash products, and Morita equivalence”. In: Journal of Algebra 133.2 (1990), pp. 351–372. [CG11] S. Caenepeel and I. Goyvaerts. “Monoidal Hom-Hopf algeb ras”. In: Com- munications in Algebra 39.6 (2011), pp. 2216–2240. [CMZ02] S. Caenepe...

work page 1990
[13]

Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras

EMS Tracts in Math- ematics. Appendix 5 by Michael M¨ uger and Appendices 6 and 7 by Alex is Virelizier. European Mathematical Society (EMS), Z¨ urich, 2010, pp. xiv+276. [Vec03] P. Vecserny´ es. “Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras”. In: Journal of Algebra 270.2 (2003), pp. 471–520. [VW06] A. Van Daele and S. ...

work page 2003
[27]

Metric spaces, generalized logic, and close d categories

[Law73] F. W. Lawvere. “Metric spaces, generalized logic, and close d categories”. In: Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135–166 (1974). [LS69] R. G. Larson and M. E. Sweedler. “An associative orthogona l bilinear form for Hopf algebras”. In: American Journal of Mathematics 91 (1969), pp. 75–

work page 1973
[94]

When Hopf algebras are Frobenius algebras

[Par71] B. Pareigis. “When Hopf algebras are Frobenius algebras”. In: Journal of Algebra 18 (1971), pp. 588–596. [Sar20] P. Saracco. “Antipodes, preantipodes and Frobenius fun ctors”. In: Journal of Algebra and Its Applications (2020). to appear. [Sch96] P. Schauenburg. “Hopf bi-Galois extensions”. In: Comm. Algebra 24.12 (1996), pp. 3797–3825. [Som02] Y....

work page 1971
[235]

Monoidal bicategories and Hopf algeb roids

[DS97] B. Day and R. Street. “Monoidal bicategories and Hopf algeb roids”. In: Advances in Mathematics 129.1 (1997), pp. 99–157. [IK10] M. C. Iovanov and L. Kadison. “When weak Hopf algebras are Frobenius”. In: Proceedings of the American Mathematical Society 138.3 (2010), pp. 837–

work page 1997
[845]

Braided tensor categories

[JS93] A. Joyal and R. Street. “Braided tensor categories”. In: Advances in Math- ematics 102.1 (1993), pp. 20–78. [Kel05] G. M. Kelly. “Basic concepts of enriched category theory”. In: Reprints in Theory and Applications of Categories 10 (2005), pp. vi+137. [KV18] B.-J. Kahng and A. Van Daele. “The Larson-Sweedler theore m for weak multiplier Hopf algebr...

work page 1993
[1787]

Topological conformal ﬁeld theories and Calab i–Yau cate- gories

Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002, pp. xiv+354. [Cos07] K. Costello. “Topological conformal ﬁeld theories and Calab i–Yau cate- gories”. In: Advances in Mathematics 210.1 (2007), pp. 165–214. REFERENCES 53 [CVW04] S. Caenepeel, J. Vercruysse, and S. Wang. “Morita theor y for corings and cleft entwining structures”. In: Journal of...

work page 2007

Show all 13 references

[1789]

Frobenius algebras and monoidal categories

Lecture Notes in Mathematics. Springer-Verlag, Berlin , 2002, pp. iv+158. [Str04a] R. Street. “Frobenius algebras and monoidal categories ”. Lecture at the Annual Meeting of the Australian Mathematical Society

2002
[2004]

Frobenius monads and pseudomonoids

[Str04b] R. Street. “Frobenius monads and pseudomonoids”. In: Journal of Mathe- matical Physics 45.10 (2004), pp. 3930–3948. [Tak99] M. Takeuchi. “Finite Hopf algebras in braided tensor catego ries”. In: Jour- nal of Pure and Applied Algebra 138.1 (1999), pp. 59–82. [Tur10] V....

2004
[2017]

Hopf comonads on naturally Frobeniu s map-monoidales

[BL16] G. B¨ ohm and S. Lack. “Hopf comonads on naturally Frobeniu s map-monoidales”. In: Journal of Pure and Applied Algebra 220.6 (2016), pp. 2177–2213. [BNS99] G. B¨ ohm, F. Nill, and K. Szlach´ anyi. “Weak Hopf Algebras I. Integral theory and C*-structure”. In: J. Algebra ...

2016
[2019]

Constru cting inﬁnite coma- trix corings from colimits

[CDV06] S. Caenepeel, E. De Groot, and J. Vercruysse. “Constru cting inﬁnite coma- trix corings from colimits”. In: Applied Categorical Structures 14.5-6 (2006), pp. 539–565. [CF94] L. Crane and I. B. Frenkel. “Four-dimensional topological q uantum ﬁeld theory, Hopf categories...

2006
[2657]

Hopf-Frobenius algebras and a Simp ler Drinfeld double

[CD19] J. Collins and R. Duncan. “Hopf-Frobenius algebras and a Simp ler Drinfeld double”. arXiv:1905.00797 [math.QA], to appear in Electronic Proceedings in Theoretical Computer Science

1905 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Oplax Hopf Algebras

[BCV16] E. Batista, S. Caenepeel, and J. Vercruysse. “Hopf Cate gories”. In: Algebras and Representation Theory 19.5 (2016), pp. 1173–1216. [BFVV17] M. Buckley, T. Fieremans, C. Vasilakopoulou, and J. Vercr uysse. “Oplax Hopf Algebras”. arXiv:1710.01465 [math.CT], to appear in Higher Struc- tures

work page Pith review arXiv 2016

[2] [10]

Hopf Galois ex tensions, smash products, and Morita equivalence

Topology and physics. 1994, pp. 5136–5154. [CFM90] M. Cohen, D. Fischman, and S. Montgomery. “Hopf Galois ex tensions, smash products, and Morita equivalence”. In: Journal of Algebra 133.2 (1990), pp. 351–372. [CG11] S. Caenepeel and I. Goyvaerts. “Monoidal Hom-Hopf algeb ras”. In: Com- munications in Algebra 39.6 (2011), pp. 2216–2240. [CMZ02] S. Caenepe...

work page 1990

[3] [13]

Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras

EMS Tracts in Math- ematics. Appendix 5 by Michael M¨ uger and Appendices 6 and 7 by Alex is Virelizier. European Mathematical Society (EMS), Z¨ urich, 2010, pp. xiv+276. [Vec03] P. Vecserny´ es. “Larson-Sweedler theorem and the role of grouplike elements in weak Hopf algebras”. In: Journal of Algebra 270.2 (2003), pp. 471–520. [VW06] A. Van Daele and S. ...

work page 2003

[4] [27]

Metric spaces, generalized logic, and close d categories

[Law73] F. W. Lawvere. “Metric spaces, generalized logic, and close d categories”. In: Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135–166 (1974). [LS69] R. G. Larson and M. E. Sweedler. “An associative orthogona l bilinear form for Hopf algebras”. In: American Journal of Mathematics 91 (1969), pp. 75–

work page 1973

[5] [94]

When Hopf algebras are Frobenius algebras

[Par71] B. Pareigis. “When Hopf algebras are Frobenius algebras”. In: Journal of Algebra 18 (1971), pp. 588–596. [Sar20] P. Saracco. “Antipodes, preantipodes and Frobenius fun ctors”. In: Journal of Algebra and Its Applications (2020). to appear. [Sch96] P. Schauenburg. “Hopf bi-Galois extensions”. In: Comm. Algebra 24.12 (1996), pp. 3797–3825. [Som02] Y....

work page 1971

[6] [235]

Monoidal bicategories and Hopf algeb roids

[DS97] B. Day and R. Street. “Monoidal bicategories and Hopf algeb roids”. In: Advances in Mathematics 129.1 (1997), pp. 99–157. [IK10] M. C. Iovanov and L. Kadison. “When weak Hopf algebras are Frobenius”. In: Proceedings of the American Mathematical Society 138.3 (2010), pp. 837–

work page 1997

[7] [845]

Braided tensor categories

[JS93] A. Joyal and R. Street. “Braided tensor categories”. In: Advances in Math- ematics 102.1 (1993), pp. 20–78. [Kel05] G. M. Kelly. “Basic concepts of enriched category theory”. In: Reprints in Theory and Applications of Categories 10 (2005), pp. vi+137. [KV18] B.-J. Kahng and A. Van Daele. “The Larson-Sweedler theore m for weak multiplier Hopf algebr...

work page 1993

[8] [1787]

Topological conformal ﬁeld theories and Calab i–Yau cate- gories

Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2002, pp. xiv+354. [Cos07] K. Costello. “Topological conformal ﬁeld theories and Calab i–Yau cate- gories”. In: Advances in Mathematics 210.1 (2007), pp. 165–214. REFERENCES 53 [CVW04] S. Caenepeel, J. Vercruysse, and S. Wang. “Morita theor y for corings and cleft entwining structures”. In: Journal of...

work page 2007