A locally finite category is defined as a category in which every arrow admits only finitely many
different ways to be factorized by composable arrows. The large algebra of such categories over
some fields may be defined, and with it a group of invertible series (under multiplication). For
certain particular locally finite categories, a substitution operation, generalizing the usual substitution
of formal power series, may be defined, and with it a group of reversible series (invertible
under substitution). Moreover, both groups are actually affine groups. In this contribution, we
introduce their coordinate Hopf algebras which are both free as commutative algebras. The
semidirect product structure obtained from the action of reversible series on invertible series by
anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras
under the form of a smash coproduct.
On commutative algebras of unbounded operators