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We consider three a priori totally different setups for Hopf algebras fromnumber theory, mathematical physics and algebraic topology. These are the Hopfalgebra of Goncharov for multiple zeta values, that of Connes-Kreimer forrenormalization, and a Hopf algebra constructed by Baues to study double loopspaces. We show that these examples can be successively unified by consideringsimplicial objects, co-operads with multiplication and Feynman categories atthe ultimate level. These considerations open the door to new constructions andreinterpretations of known constructions in a large common framework, which ispresented step-by-step with examples throughout. In this first part of twopapers, we concentrate on the simplicial and operadic aspects.

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