Three Hopf algebras from number theory, physics &amp; topology, and their common background I: operadic &amp; simplicial aspects

Gálvez-Carrillo, Imma; Kaufmann, Ralph M.; Tonks, Andrew

doi:10.4310/cntp.2020.v14.n1.a1

Cited by 10 publications

(59 citation statements)

References 33 publications

(79 reference statements)

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“…More generally for a Feynman category we will use deconcatenation and monoidal product to construct a bi-algebra and then show that under certain natural conditions a quotient of this bi-algebra yields a Hopf algebras. This makes the theory particularly transparent and allows to recover the previous constructions of [GCKT20] by specialization. Or, taking the reverse perspective, we can generalize the constructions appearing in [GCKT20] by lifting them to the categorical level.…”

Section: The General Case: Bi-and Hopf Algebras Frommentioning

confidence: 99%

“…This makes the theory particularly transparent and allows to recover the previous constructions of [GCKT20] by specialization. Or, taking the reverse perspective, we can generalize the constructions appearing in [GCKT20] by lifting them to the categorical level.…”

Section: The General Case: Bi-and Hopf Algebras Frommentioning

confidence: 99%

“…The natural setup for this definitive source of Hopf algebras of this type are the Feynman categories of [KW17]. The results of [GCKT20] can be re-derived by restricting to special types of Feynman categories. As in the first part, the key observation is that the Hopf algebras are quotients of bi-algebras.…”

Section: Introductionmentioning

confidence: 99%

“…The case relevant for co-operads with multiplication, treated in the first part, is the Feynman category of finite sets and surjections and its enrichments by operads. The constructions of the bi-algebra then correspond to the pointed free case considered in [GCKT20] if the co-operad is the dual of an operad. Invoking opposite categories, one can treat the co-operads appearing in [GCKT20] directly.…”

Section: Introductionmentioning

confidence: 99%

“…These constructions also give interrelations between the examples. Among the examples discussed are the examples of [GCKT20] which are analyzed in various contexts that provide new depths of understanding. In particular, we revisit operads, simplicial structures and Joyal duality in the context of Feynman categories, decorations and enrichments.…”

Section: Introductionmentioning

confidence: 99%

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Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

Gálvez-Carrillo

Kaufmann

Tonks

2020

Communications in Number Theory and Physics

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We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation. arXiv:2001.08722v1 [math.AT]

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Section: The General Case: Bi-and Hopf Algebras Frommentioning

confidence: 99%

Section: The General Case: Bi-and Hopf Algebras Frommentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

Gálvez-Carrillo

Kaufmann

Tonks

2020

Communications in Number Theory and Physics

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Adams’ cobar construction as a monoidal -coalgebra model of the based loop space

Medina-Mardones,

Rivera

2024

Forum of Mathematics, Sigma

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We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty $ -coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.

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Feynman categories and representation theory

Kaufmann¹

2021

Representations of Algebras, Geometry And Physics

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We give a presentation of Feynman categories from a representation-theoretical viewpoint.Feynman categories are a special type of monoidal categories and their representations are monoidal functors. They can be viewed as a far reaching generalization of groups, algebras and modules. Taking a new algebraic approach, we provide more examples and more details for several key constructions. This leads to new applications and results.The text is intended to be a self-contained basis for a crossover of more elevated constructions and results in the fields of representation theory and Feynman categories, whose applications so far include number theory, geometry, topology and physics.Organization of the text. The text is designed to be as self-contained as possible and is aimed at a diverse audience.We start in §1 with collecting classic results for groups and quiver representations, but reformulated in categorical language. This presentation might be of independent interest as a primer.The next paragraph, §2, contains the definition of Feynman category introduced as a special type of monoidal category. The representations are then given by strong monoidal functors. The development is parallel to that of §1. The presentation of indexed Feynman categories is new. The section ends with examples which are based on finite sets. Here we provide new details. The representations of these are various kinds of algebras. The group(oid) representations are also included as a basic example. Further examples are provided by graphical Feynman categories. The theory of graphs we use is detailed in Appendix A.We then turn to various constructions for Feynman categories in §3. These yield Feynman categories whose representations are lax-monoidal or simply functors. At this level the finite set based Feynman categories have FI-modules and (co)-(semi)-simplicial objects as objects. The next operation is that of decoration. It yields the graphical Feynman categories that encode operadic-types, see Table 7. The next construction is the plus construction. Here we give a detailed exposition of the condensed presentation in [KW17, §3.6], providing several explicit calculations. The new precision yields gcp-version of the plus-construction, which is a generalization of hyper version contained in [KW17, §3.7]. The relationship to indexing is also made more explicit here then previously. A detailed graphical based analysis is given in Appendix B, where we also give a careful discussion of levels.In §4 we tackle the enriched version. This is technically the most demanding and contains many new details. The bar/co-bar transformation and a dual transformation, aka. Feynman transform along with the master equations are discussed in §5. Traditionally the bar/co-bar adjunction can be used to define resolutions. For this one needs a model structure in general. The relevant details are reviewed in Appendix D. The W-construction is reviewed in §6. Here we also reconstruct the associahedra in their cubical decompositions.We end the paper with an outlook, §7...

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Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Cited by 10 publications

References 33 publications

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

Adams’ cobar construction as a monoidal -coalgebra model of the based loop space

Feynman categories and representation theory

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