Lectures on Feynman Categories

Kaufmann, Ralph M.

doi:10.1007/978-3-319-72299-3_19

Cited by 18 publications

(34 citation statements)

References 29 publications

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“…We call a Feynman category strict if the monoidal structure on F is strict, ι is an inclusion, and V ⊗ = Iso(F) where we insist on using the strict free monoidal category, see e.g. [Kau17] for a thorough discussion. Up to equivalence in V, F and in F this can always be achieved.…”

Section: Feynman Categoriesmentioning

confidence: 99%

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

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 algebras 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, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

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Section: Feynman Categoriesmentioning

confidence: 99%

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

Gálvez-Carrillo

Kaufmann

Tonks

2020

Communications in Number Theory and Physics

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“…This leads to a category of elements construction for symmetric lax (co)monoidal functors. Some steps in this direction are made in [24 [15] as tools to understand moduli spaces of surfaces and algebraic curves. Since their introduction they have proved useful in other areas of mathematics as well, e.g.…”

Section: The Resulting Equivalence Of Categories El O (A)-algmentioning

confidence: 99%

“…We arrive at the following reformulation of the definition of a Feynman category of [24]: A Feynman category is a framed symmetric monoidal category (F, V, ι) which is hereditarily framed in the sense that the double slice category F ↓ F itself is a framed symmetric monoidal category with respect to the "groupoid" (F ↓ V) iso , i.e. the canonical map (F ↓ V) ⊗ iso → (F ↓ F) iso is an equivalence of symmetric monoidal categories.…”

Section: Feynman Categories and Multicategoriesmentioning

confidence: 99%

“…the canonical map (F ↓ V) ⊗ iso → (F ↓ F) iso is an equivalence of symmetric monoidal categories. Notice that our smallness condition (V small) is slightly more restrictive than the one used in [24]. An algebra for a Feynman category F or, as we shall say, an F-operad is a strong symmetric monoidal functor F → (Set, ×, Set ).…”

Section: Feynman Categories and Multicategoriesmentioning

confidence: 99%

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Comprehensive factorisation systems

Berger¹,

Kaufmann²

2017

Tbilisi Math. J.

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We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems exist for the categories of topological spaces, simplicial sets, small multicategories and Feynman categories. In each case comprehensive factorisation induces a natural notion of universal covering, leading to a Galois-type definition of fundamental group for based objects of the category.

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“…This category is sometimes also denoted by Σ and it is the skeleton of Iso(FinSet), where FinSet is the category of finite sets with set maps. For more details, see [Kau17], especially §2.4. Consider a strict Feynman category F = ( * , F, ı) with Obj(F) = N 0 .…”

Section: Constructions and Examplesmentioning

confidence: 99%

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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Lectures on Feynman Categories

Cited by 18 publications

References 29 publications

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

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

Comprehensive factorisation systems

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

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