Integrals along bimonoid homomorphisms

Minkyu Kim

REVIEW 2 major objections 5 minor 22 references

Integrals along bimonoid homomorphisms

T0 review · 2 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 This paper proves that a normalized generator integral along a Hopf monoid homomorphism exists exactly when its kernel has a normalized integral and its cokernel has a normalized cointegral, and that when it exists the integral is unique.

desk verdict A serious categorical extension of Hopf algebra integrals with a clean kernel/cokernel characterization; limited by strong assumptions and unverifiable string-diagram proofs, but no load-bearing flaw is apparent. read the letter →

arxiv 1908.01658 v7 pith:CJH5NFTQ submitted 2019-08-05 math.QA

Minkyu Kim This is my paper · ORCID

classification math.QA MSC 16T0518M0518E10

keywords HopfmonoidsintegralsalonghomomorphismscointegralsbicommutativebimonoidssymmetricmonoidalcategoriesvolumeonabelianFredholmindexnormalizedgeneratorintegral

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

Homomorphisms between Hopf monoids generalize group homomorphisms, and integrals generalize Haar measures. This paper introduces an integral along a bimonoid homomorphism—a morphism from the target back to the source satisfying averaging axioms—and proves a simple existence criterion: under the paper's assumptions, a normalized generator integral exists exactly when the kernel Hopf monoid has a normalized integral and the cokernel Hopf monoid has a normalized cointegral, and then it is unique. This unifies the classical integral and cointegral of a bimonoid as special cases, with the integral along the counit being an integral and the integral along the unit being a cointegral. A sympathetic reader should care because the result turns a delicate existence question about operators into a checkable condition on two associated objects, and because the same framework yields a notion of volume and a Fredholm index for Hopf monoids that the author uses to build TQFTs.

What carries the argument

The load-bearing mechanism is the normalized generator integral along $\xi$: a morphism $\mu : B \to A$ satisfying left and right integral axioms and the normalization $\xi \circ \mu \circ \xi = \xi$, which generates all integrals by acting through endomorphisms of the unit object. Around this, the paper builds two devices. First, a bimonoid is small (cosmall) when, for every action (coaction), the canonical map from the invariant object to the stabilized object is an isomorphism; Theorem 6.13 identifies smallness with having a normalized integral when every idempotent splits. Second, the monoidal structure of $C$ is bistable when the tensor product preserves the equalizers and coequalizers used to define stabilized objects; this makes every homomorphism between bicommutative bimonoids binormal, so it has well-defined kernel and cokernel Hopf monoids and a coimage-image isomorphism. The integral is then the composition of normalized integrals attached to the kernel and cokernel maps.

What would settle it

A concrete check is the group-algebra case. Take finite groups $G,H$ and a homomorphism $\rho : G \to H$ whose finite kernel has order divisible by the characteristic of $k$; the paper's formula for the normalized integral is $\mu(h) = |\mathrm{Ker}(\rho)|^{-1} \sum_{\rho(g)=h} g$, so the denominator is not invertible and the criterion predicts no normalized generator integral along $k\rho$. To falsify Theorem 1.1 outright, one would need a category satisfying Assumptions 0–2 with a homomorphism whose kernel has a normalized integral and cokernel a normalized cointegral but no normalized generator integral, or two distinct normalized integrals along the same homomorphism.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for bicommutative Hopf monoids $A$ and $B$ in a symmetric monoidal category satisfying Assumptions 0–2, a Hopf homomorphism $\xi : A \to B$ admits a normalized generator integral $\mu : B \to A$ if and only if $\mathrm{Ker}(\xi)$ has a normalized integral and $\mathrm{Cok}(\xi)$ has a normalized cointegral; when it exists, $\mu$ is unique. The 'if' direction is constructive: the integral is assembled from the normalized integral of the kernel and the normalized cointegral of the cokernel, together with an isomorphism between the coimage and image of $\xi$ that the binormal decomposition provides. The characterization is obtained by showing that, under the assumptions, the existence of such an integral is equivalent to the kernel being 'small' and the cokernel being 'cosmall,' which in turn is equivalent to having the relevant normalized (co)integrals when idempotents split. The paper then packages the multiplicative behaviour of these integrals into a normalized 2-cocycle and shows it is a coboundary, yielding a strictly functorial assignment of integrals to Fredholm homomorphisms.

Load-bearing premise

The proof depends on the assumption that the monoidal product preserves the equalizers and coequalizers used to build kernels and cokernels (bistability); if that fails, those kernel and cokernel Hopf monoids may not exist, and the theorem's conditions are undefined.

Editorial extensions

If this is right

In the group-algebra example, a homomorphism $\rho : G \to H$ admits a normalized generator integral along $k\rho$ exactly when $\mathrm{Ker}(\rho)$ and $\mathrm{Cok}(\rho)$ are finite with orders coprime to the characteristic of $k$, and the integral is $\mu(h) = |\mathrm{Ker}(\rho)|^{-1}\sum_{\rho(g)=h} g$.
Every weakly well-decomposable, weakly pre-Fredholm homomorphism has a unique normalized generator integral, and the associated endomorphisms $\mu \circ \xi$ and $\xi \circ \mu$ are idempotents on the source and target (Corollary 7.2 and Lemma 9.5).
The inverse volume is a volume on the abelian category of bismall bicommutative Hopf monoids: it is multiplicative under short exact sequences, tensor products, and duals (Theorem 15.7 and Proposition 11.4).
Fredholm homomorphisms between bicommutative Hopf monoids form a category, and the Fredholm index is multiplicative under composition and invariant under finite perturbations (Proposition 15.9).
The normalized 2-cocycle $\omega_C$ is a coboundary, so the assignment $\xi \mapsto \xi_!$ is strictly functorial on the Fredholm category (Propositions 15.13 and 15.16).

Reading between the lines

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

One extension beyond the paper: the criterion may survive without bicommutativity whenever the homomorphism is binormal in the relevant sense; the group-algebra example with arbitrary (non-abelian) groups points in that direction.
Read as a categorical Euler characteristic, the inverse volume and its Fredholm index suggest an index theorem for bimonoid homomorphisms: the index should be invariant under suitable homotopies, a question the paper leaves open.
The strict functoriality of the integral assignment $\xi \mapsto \xi_!$ is a choice-dependent splitting of the 2-cocycle $\omega_C$; different splittings should yield cohomologous assignments, which in the planned TQFT may correspond to different normalizations of the same path integral.

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

Colleague,

Bottom line: this is a genuinely useful categorical paper, not a repackaging. It introduces an integral along a bimonoid homomorphism, shows it simultaneously generalizes ordinary integrals and cointegrals (via counit and unit), and proves a real characterization: for bicommutative Hopf monoids, a normalized generator integral along a Hopf homomorphism exists iff the kernel has a normalized integral and the cokernel has a normalized cointegral. That is Theorem 1.1, and it is substantive. The concrete group-algebra example (Example 3.6, later Example 9.12) makes the content recognizable.

What the paper does well: the proof structure is explicit. Uniqueness is proven in Proposition 7.1, necessity in Theorem 7.5, and sufficiency via a concrete construction in Theorem 9.9. Assumptions are declared, not hidden. The inverse volume is shown to satisfy the volume axioms, not assumed into existence. The Fredholm-index analogue is honest as an application and a preliminary to the author's promised TQFT construction. Citation pattern looks fair; the references to Bespalov et al. and Larson-Sweedler are relevant, and the self-reference to the sequel is clearly labeled.

Now the soft spots, in proportion. The main theorem is explicitly conditional on Assumptions 0-2. Assumption 1 (bistability of the monoidal structure) and Assumption 2 (abelianness of Hopfbc(C)) are strong, and they are verified only for Vec^b_k. That is a scope restriction, not an internal contradiction, but it limits how broadly the result can be applied as it stands. The bigger practical concern is that central proofs, especially Lemma 9.1 and Theorem 9.9, rely heavily on string-diagram figures rather than written equations. In this rendering those figures cannot be checked, so a referee cannot quickly certify the key compatibility checks. I would ask the author to convert the nontrivial diagram arguments into algebraic text or a formalized proof, not because I suspect error, but because verifiability is currently bottlenecked there.

I agree with the stress-test note: no circularity, no hidden assumption beyond those stated. The theorem holds up as a conditional statement.

Who is this for? Researchers working on Hopf monoids in tensor categories, integrals, and state-sum TQFT foundations. It deserves a serious referee rather than a desk rejection. I would send it to review with an instruction to focus on Lemma 9.1 and Theorem 9.9, and to check that Assumptions 1-2 are not so restrictive that the advertised applications lose force.

Referee Report

2 major / 5 minor

Summary. The paper introduces a notion of an integral along a bimonoid homomorphism, generalizing both integrals and cointegrals of bimonoids, and defines a normalized generator integral. Its main theorem (Theorem 1.1) characterizes, under three explicit assumptions on the ambient symmetric monoidal category C, the existence of a normalized generator integral along a Hopf homomorphism between bicommutative Hopf monoids in terms of existence of a normalized integral on the kernel and a normalized cointegral on the cokernel, with uniqueness. The proof proceeds by establishing binormality of homomorphisms under bistability, relating smallness to normalized integrals, proving necessity and sufficiency through the weakly well-decomposable / weakly pre-Fredholm construction of μξ, and then deriving the main theorem as Corollary 9.11. The paper then develops inverse volume, applies it to define Fredholm homomorphisms and a Fredholm index in the category of bicommutative Hopf monoids, and constructs a functorial integral assignment from a normalized 2-cocycle.

Significance. If the results are correct, this is a substantial contribution to the integral theory of bimonoids. The paper provides a concrete, explicit construction of integrals along homomorphisms, gives a clean kernel/cokernel criterion, and proves uniqueness under mild hypotheses. The inverse volume and Fredholm index formalism are natural and potentially useful for the announced TQFT applications. I credit the paper for stating its assumptions clearly (Assumptions 0-2), for verifying them for the category Vec^b_k, for explicitly constructing μξ rather than assuming its existence, and for including a detailed proof architecture with stated intermediate results. The main limitation is that the applicability beyond Vec^b_k rests on strong assumptions, in particular bistability of the monoidal structure and abelianness of Hopfbc(C), which are not established for other tensor categories; this is a scope restriction rather than an internal inconsistency.

major comments (2)

[Section 5, Proposition 5.8] The proof that every homomorphism between bicommutative bimonoids is binormal is load-bearing, and its key step is the claim that the triple (A, α^→_ξ, B) is a bicommutative bimonoid in Act_l(C). This claim is supported only by diagrams (45)-(48), with the text noting that commutativity of B is used. Since the entire reduction of the main theorem to Corollary 9.11 depends on this proposition, the action-compatibility equations for ∇, Δ, η, ε should be written out explicitly in the text, or the relevant diagrams should be fully explained, so that the reader can verify that no hidden cocommutativity or non-commutativity assumption is needed.
[Section 9, Lemma 9.1 and Theorem 9.9] The central sufficiency proof is delegated to Figures 20-26, which are referenced but not reproduced in the text. Theorem 9.9 is the engine behind Corollary 9.11 and hence Theorem 1.1, so the correctness of the diagram chases is essential. In the published version, all such figures must be present and legible, and ideally the key computations in Lemma 9.1 and Theorem 9.9 should be accompanied by at least one fully written equational chase; otherwise the proof is not independently verifiable from the text alone.

minor comments (5)

[Throughout] There are numerous typographical errors, including 'coequazliers', 'equazliers', 'normlaized', 'bimnoid', 'the te normalized', and 'JPSJ Grant-in-Aid' (presumably JSPS). These should be corrected in a final revision.
[Example 11.3] The text uses '7G' and 'p7 Gq' where the order of the group is intended; this should read |G| consistently.
[Section 15, Assumptions 0-2] The abstract and introduction state Theorem 1.1 with a parenthetical reference to Assumptions 0, 1, 2, but the assumptions themselves are only formally introduced in Section 15. A forward reference in the introduction or an early statement of the assumptions would improve readability.
[Definition 3.11 and Proposition 3.12] The definition of a generator integral is given for both left and right integrals, but Proposition 3.12 only proves the equivalence for integrals along the counit in one direction and leaves the other to the reader; a brief indication of the dual argument would be helpful.
[Theorem 12.1 and Corollary 12.4] The scalar λ in Theorem 12.1 is identified with ⟨cok(ξ) ∘ ker(ξ')⟩, but this identification is stated after the proof of Corollary 12.4; moving it into the statement of Theorem 12.1 would make the result more transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is a proven conditional characterization under explicit Assumptions 0-2; all equivalences are established by proof, not by definition.

full rationale

I traced the derivation chain of Theorem 1.1 and Corollary 9.11. The notion of an integral along a bimonoid homomorphism (Definition 3.4) is axiomatic, and the notions of normalized (co)integral (Definition 3.1), smallness/cosmallness (Definition 6.2), and weak pre-Fredholmness (Definition 9.7) are distinct and are connected by proved theorems rather than identified by fiat. In particular, Theorem 6.13 proves the equivalence between smallness and existence of a normalized integral under the split-idempotent assumption, and Corollary 9.11 derives the main existence criterion from Theorem 6.13, Theorem 7.5/7.6, and Corollary 9.10. The sufficiency direction constructs the normalized generator integral explicitly from normalized integrals on the kernel and cointegral on the cokernel; the necessity direction derives those integrals from a given normalized integral. Neither direction assumes the conclusion. The volume vol^{-1} is defined as the composition sigma_A-bar composed with sigma_A, and Theorem 15.7 verifies the exact-sequence axiom rather than assuming it; the Fredholm index is introduced as an analogue, and its multiplicativity is proved in Lemma 14.7. The only self-citation, [10], concerns future TQFT applications and is not load-bearing in any proof. Assumptions 0-2 are stated explicitly, and their verification for Vec^b_k is supplied by Proposition 4.6/Example 4.7 plus external references [17,20]. No equation is equal to its own input by construction, and no fitted parameter is relabeled as a prediction. The derivation is therefore self-contained and non-circular.

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

The paper is definitional and theorem-based with no numerical fitting. Its conclusions depend on the structural assumptions on C stated in Section 15: existence of equalizers and coequalizers, bistable monoidal structure, and abelianity of the category of bicommutative Hopf monoids. These are domain assumptions rather than standard mathematical axioms, and they restrict the proven scope, with Vec^b_k as the primary example.

assumptions (4)

standard math C is a symmetric monoidal category with coherence isomorphisms (associator, unitors, symmetry).
Throughout the paper as the ambient setting; standard categorical background from Mac Lane and Aguiar-Mahajan.
domain assumption Assumption 0: C has arbitrary equalizers and coequalizers.
Section 15; needed for stabilized objects, kernels and cokernels, and to ensure every idempotent splits via Proposition 6.6.
domain assumption Assumption 1: the monoidal structure of C is bistable (Definition 4.4).
Section 15; used in Proposition 5.8 to prove every homomorphism between bicommutative bimonoids is binormal, a key step toward well-decomposability.
domain assumption Assumption 2: the category Hopf^bc(C) of bicommutative Hopf monoids is abelian.
Section 15; needed for the volume on an abelian category, Fredholm index, and exact sequence arguments. For Vec^b_k this is cited to Takeuchi and Newman, but no other examples are given.

invented entities (4)

Integral along a bimonoid homomorphism (Definition 3.4)
purpose: Simultaneously generalizes integral and cointegral of bimonoids; the central object of study.
Definitional; its content is validated by the characterization theorem rather than by external empirical data.
Normalized generator integral (Definition 3.11)
purpose: Selects a canonical integral along a homomorphism that generates all other integrals.
Definitional; used to obtain uniqueness and the bijection with End_C(1) in Theorem 8.5.
Volume on an abelian category (Definition 14.1)
purpose: Generalizes dimension of vector spaces and order of abelian groups to a multiplicative invariant for short exact sequences.
Definitional; provides the basis for Fredholm index theory in this categorical setting.
Fredholm homomorphism between bicommutative Hopf monoids (Definition 15.8)
purpose: Analogue of a Fredholm operator, with an index computed from the inverse volumes of kernel and cokernel.
Definitional; built on the constructed volume and used to produce functorial integrals.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Integrals along bimonoid homomorphisms." pith.science (2026). https://pith.science/paper/CJH5NFTQ

@misc{pith2026190801658,
  author       = {Pith},
  title        = {Pith review of: Integrals along bimonoid homomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CJH5NFTQ}},
  note         = {Machine review of arXiv:1908.01658}
}

read the original abstract

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a normalized generator integral, along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism. We introduce a notion of a volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFTs.

Figures

Figures reproduced from arXiv: 1908.01658 by the authors.

**Figure 21.** Thus, we obtain µ0 P Intrpπq [PITH_FULL_IMAGE:figures/full_fig_p028_21.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 19 canonical work pages

[1]

Monoidal functors, species and Hopf algebras , vol- ume 29

Marcelo Aguiar and Swapneel Arvind Mahajan. Monoidal functors, species and Hopf algebras , vol- ume 29. American Mathematical Society Providence, RI, 2010

work page 2010
[2]

Kitaev’s lattice model and Turaev-Viro TQFTs

Benjamin Balsam and Alexander Kirillov Jr. Kitaev’s lattice model and Turaev-Viro TQFTs. arXiv preprint arXiv:1206.2308, 2012

arXiv 2012
[3]

Invariants of piecewise-linear 3-manifolds.Transactions of the American Mathematical Society, 348(10):3997–4022, 1996

John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds.Transactions of the American Mathematical Society, 348(10):3997–4022, 1996

work page 1996
[4]

Integrals for braided Hopf algebras

Yuri Bespalov, Thomas Kerler, V olodymyr Lyubashenko, and Vladimir Turaev. Integrals for braided Hopf algebras. Journal of Pure and Applied Algebra, 148(2):113–164, 2000

work page 2000
[5]

A hierarchy of topological tensor network states

Oliver Buerschaper, Juan Mart ´ın Mombelli, Matthias Christandl, and Miguel Aguado. A hierarchy of topological tensor network states. Journal of Mathematical Physics, 54(1):012201, 2013

work page 2013
[6]

Topological gauge theories and group cohomology

Robbert Dijkgraaf and Edward Witten. Topological gauge theories and group cohomology. Communica- tions in Mathematical Physics, 129(2):393–429, 1990

work page 1990
[7]

Chern-Simons theory with ﬁnite gauge group

Daniel S Freed and Frank Quinn. Chern-Simons theory with ﬁnite gauge group. Communications in Math- ematical Physics, 156(3):435–472, 1993

work page 1993
[8]

Perturbation theory for linear operators , volume 132

Tosio Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013

2013

Show all 22 references

[9]

Non-semisimple topological quantum ﬁeld theories for 3-manifolds with corners, volume 1765

Thomas Kerler and V olodymyr V Lyubashenko. Non-semisimple topological quantum ﬁeld theories for 3-manifolds with corners, volume 1765. Springer Science & Business Media, 2001

2001
[10]

A family of TQFT’s associated with homology theory

Minkyu Kim. A family of TQFT’s associated with homology theory. arXiv preprint arXiv:2006.10438 , 2020

2006 arXiv
[11]

Involutory Hopf algebras and 3-manifold invariants

Greg Kuperberg. Involutory Hopf algebras and 3-manifold invariants. International Journal of Mathemat- ics, 2(01):41–66, 1991

1991
[12]

Non-involutory Hopf algebras and 3-manifold invariants

Greg Kuperberg. Non-involutory Hopf algebras and 3-manifold invariants. arXiv preprint q-alg/9712047, 1997

1997 arXiv
[13]

An associative orthogonal bilinear form for Hopf algebras

Richard Gustavus Larson and Moss Eisenberg Sweedler. An associative orthogonal bilinear form for Hopf algebras. American Journal of Mathematics, 91(1):75–94, 1969

1969
[14]

Categories for the working mathematician, volume 5

Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013

2013
[15]

Kitaev lattice models as a Hopf algebra gauge theory

Catherine Meusburger. Kitaev lattice models as a Hopf algebra gauge theory. Communications in Mathe- matical Physics, 353(1):413–468, 2017

2017
[16]

On the structure of Hopf algebras

John W Milnor and John C Moore. On the structure of Hopf algebras. Annals of Mathematics , pages 211–264, 1965

1965
[17]

A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras

Kenneth Newman. A correspondence between bi-ideals and sub-Hopf algebras in cocommutative Hopf algebras. Journal of Algebra, 36(1):1–15, 1975

1975
[18]

The order of the antipode of a ﬁnite dimensional Hopf algebra is ﬁnite.American Journal of Mathematics, pages 333–355, 1976

David E Radford. The order of the antipode of a ﬁnite dimensional Hopf algebra is ﬁnite.American Journal of Mathematics, pages 333–355, 1976

1976
[19]

Integrals for Hopf algebras

Moss Eisenberg Sweedler. Integrals for Hopf algebras. Annals of Mathematics, pages 323–335, 1969

1969
[20]

A correspondence between Hopf ideals and sub-Hopf algebras

Mitsuhiro Takeuchi. A correspondence between Hopf ideals and sub-Hopf algebras. manuscripta mathe- matica, 7(3):251–270, 1972

1972
[21]

State sum invariants of 3-manifolds and quantum 6j-symbols.Topol- ogy, 31(4):865–902, 1992

Vladimir G Turaev and Oleg Ya Viro. State sum invariants of 3-manifolds and quantum 6j-symbols.Topol- ogy, 31(4):865–902, 1992

1992
[22]

On Dijkgraaf-Witten invariant for 3-manifolds

Michihisa Wakui. On Dijkgraaf-Witten invariant for 3-manifolds. Osaka Journal of Mathematics , 29(4):675–696, 1992. Graduate School of MathematicalSciences, University of Tokyo Email address: kim@ms.u-tokyo.ac.jp

1992

Pith tools

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

[1] [1]

Monoidal functors, species and Hopf algebras , vol- ume 29

Marcelo Aguiar and Swapneel Arvind Mahajan. Monoidal functors, species and Hopf algebras , vol- ume 29. American Mathematical Society Providence, RI, 2010

work page 2010

[2] [2]

Kitaev’s lattice model and Turaev-Viro TQFTs

Benjamin Balsam and Alexander Kirillov Jr. Kitaev’s lattice model and Turaev-Viro TQFTs. arXiv preprint arXiv:1206.2308, 2012

arXiv 2012

[3] [3]

Invariants of piecewise-linear 3-manifolds.Transactions of the American Mathematical Society, 348(10):3997–4022, 1996

John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds.Transactions of the American Mathematical Society, 348(10):3997–4022, 1996

work page 1996

[4] [4]

Integrals for braided Hopf algebras

Yuri Bespalov, Thomas Kerler, V olodymyr Lyubashenko, and Vladimir Turaev. Integrals for braided Hopf algebras. Journal of Pure and Applied Algebra, 148(2):113–164, 2000

work page 2000

[5] [5]

A hierarchy of topological tensor network states

Oliver Buerschaper, Juan Mart ´ın Mombelli, Matthias Christandl, and Miguel Aguado. A hierarchy of topological tensor network states. Journal of Mathematical Physics, 54(1):012201, 2013

work page 2013

[6] [6]

Topological gauge theories and group cohomology

Robbert Dijkgraaf and Edward Witten. Topological gauge theories and group cohomology. Communica- tions in Mathematical Physics, 129(2):393–429, 1990

work page 1990

[7] [7]

Chern-Simons theory with ﬁnite gauge group

Daniel S Freed and Frank Quinn. Chern-Simons theory with ﬁnite gauge group. Communications in Math- ematical Physics, 156(3):435–472, 1993

work page 1993

[8] [8]

Perturbation theory for linear operators , volume 132

Tosio Kato. Perturbation theory for linear operators , volume 132. Springer Science & Business Media, 2013

2013