Connes—Kreimer—Feynman type bialgebras

Bialgebras built on the model of Connes and Kreimer are ubiquitous in mathematics. One explanation is that they stem from a categorical construction using Feynman categories. This opens bridges between combinatorics, algebra and topology as we explain. Examples include (compactifications of) moduli spaces of Riemann surfaces on one end, and inclusion/exclusion algebras on the other. In between, these are factorization systems. Using this framework, statements that things that seem analogous or similar can be made rigorous for instance in terms of functors.