This paper traces a straight line from classical Mobius 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 Mobius inversion.

中文翻译：

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从莫比乌斯反演到重整化

本文追溯了一条从经典莫比乌斯反演到 Hopf 代数微扰重整化的直线。这条线是合乎逻辑的但不完全是历史的，它只包含几个主要的抽象步骤，以及一些为了数学乐趣而详述的中间步骤。这篇论文主要是说明性的，但包含许多关于众所周知的结果的新观点。例如，Bogoliubov 递归和 Atkinson 公式之间的等价性被展示为 Weisner--Rota 递归和 Hall--Leroux 莫比乌斯反演公式之间的等价性的直接推广。