This paper introduces and develops M\"obius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and M\"obius functions is classical, we go further by establishing a direct connection between poset topology and M\"obius inversions. In particular, we show that M\"obius homology categorifies the M\"obius inversion, as its Euler characteristic coincides with the M\"obius inversion applied to the dimension function of the representation. We also present a homological version of Rota's Galois Connection Theorem, relating the M\"obius homologies of two posets connected by a Galois connection. Our main application concerns persistent homology over general posets. We prove that, under a suitable definition, the persistence diagram arises as an Euler characteristic over a poset of intervals, and thus M\"obius homology provides a categorification of the persistence diagram. This furnishes a new invariant for persistent homology over arbitrary finite posets. Finally, leveraging our homological variant of Rota's Galois Connection Theorem, we establish several results about the persistence diagram.

翻译：本文引入并发展了莫比乌斯同调，这是一种针对有限偏序集到阿贝尔范畴的表示的同调理论。尽管偏序集拓扑与莫比乌斯函数之间的联系是经典的，但我们通过建立偏序集拓扑与莫比乌斯反演之间的直接联系而更进一步。具体而言，我们证明了莫比乌斯同调将莫比乌斯反演范畴化，因为其欧拉示性数恰好等于莫比乌斯反演应用于表示的维数函数的结果。我们还提出了罗塔伽罗瓦联络定理的一个同调版本，该版本关联了由伽罗瓦联络连接的两个偏序集的莫比乌斯同调。我们的主要应用涉及一般偏序集上的持久同调。我们证明，在适当的定义下，持久图作为区间偏序集上的欧拉示性数出现，因此莫比乌斯同调为持久图提供了一个范畴化。这为任意有限偏序集上的持久同调提供了一个新的不变量。最后，利用我们罗塔伽罗瓦联络定理的同调变体，我们建立了关于持久图的若干结果。