A cogroup in a category $C$ is an object which represents a group scheme as a hom-functor on $C$, thus generalizing commutative Hopf algebras. Cogroups in groups have been introduced by D. Kan (1959) to study the homotopy theory of topological spaces with base point, and generalized to other categories by B. Eckmann and P.J. Hilton (1961-1963) to study the homology of loop spaces. With minimal standard assumptions on the category $C$, cogroups have been proved to be always free objects in C [D. Kan 1959, I. Berstein 1963, B. Fresse 1998].

Examples of such cogroups can be obtained by modifying the usual (free) Hopf algebras representing proalgebaric groups of formal series, where the category $C$ plays the role of ambient setting for the series coefficients. Formal series expanded on non-trivial combinatorial objects, like trees or Feynman graphs, gained some interest since they appeared in the renormalization procedure in quantum field theory [F. Dyson 1949, Bogoliubov-Parasiuk-Hepp-Zimmermann 1957-1969, Connes-Kreimer 2000] and recently in the study of regularity structures in stochastic differential equations [M. Hairer 2014, Bruned-Hairer-Zambotti 2016].

In order to handle the generalization to non-commutative coefficients for formal diffeomorphisms, we need to further relax groups to non-associative groups, called loops after R. Moufang (1935). In this talk I will focus on the coloop structure generalising the non-commutative Faà di Bruno Hopf algebra of formal diffeomorphisms. It is a joint work with Ivan P. Shestakov.