  This paper traces a straight line from classical M\"obius inversion to
Hopf-algebraic perturbative renormalisation. This line, which is logical but
not entirely historical, consists of just a few main abstraction steps, and
some intermediate steps dwelled upon for mathematical pleasure. The paper is
largely expository, but contains many new perspectives on well-known results.
For example, the equivalence between the Bogoliubov recursion and the Atkinson
formula is exhibited as a direct generalisation of the equivalence between the
Weisner--Rota recursion and the Hall--Leroux formula for M\"obius inversion.
