Abstract
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.
Highlights
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology
There are several Hopf algebras of Connes and Kreimer and variations of these which are of great interest in physics and number theory, e.g. [Bro17, Bro12]
The level of complexity is represented by the Connes–Kreimer Hopf algebras for renormalization defined on graphs
Summary
Given a factorization finite Feynman category F, (Biso, ⊗, η, ∆iso, iso) is a bi–algebra, both in the symmetric and the non–Σ case. For a decomposition finite (non–Σ) Feynman category the set of basic morphisms, that is objects of (F ↓ V) form a co–module, viz. A Feynman category has almost group–like identities if each of the φL and each of the φR appearing in a co–product of any idX (1.4) is an isomorphism. It is interesting to study the co–radical filtration and the ([idX], [idY ])–primitive elements in B They correspond to the generators for morphisms in Feynman categories [KW17]. This give a functor of Feynman categories enriching FS or in the planar version of FS
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