Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

A locally finite category is defined as a category in which every arrow admits only finitely manydifferent ways to be factorized by composable arrows. The large algebra of such categories oversome fields may be defined, and with it a group of invertible series (under multiplication). Forcertain particular locally finite categories, a substitution operation, generalizing the usual substitutionof formal power series, may be defined, and with it a group of reversible series (invertibleunder substitution). Moreover, both groups are actually affine groups. In this contribution, weintroduce their coordinate Hopf algebras which are both free as commutative algebras. Thesemidirect product structure obtained from the action of reversible series on invertible series byanti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebrasunder the form of a smash coproduct