M\"obius Homology

This paper introduces M\"obius homology, a homology theory for representations of finite posets into abelian categories. While the connection between poset topology and M\"obius functions is classical, we establish a direct connection between poset topology and M\"obius inversions. More precisely, the M\"obius homology categorifies the M\"obius inversion because its Euler characteristic is equal to the M\"obius inversion of the dimension function of the representation. We also introduce a homological version of Rota's Galois Connection Theorem which relates the M\"obius homology over two posets connected by a Galois connection. Our main application is to persistent homology over general posets. We show that under one definition, the persistence diagram is an Euler characteristic over a poset of intervals and hence M\"obius homology is a categorification of the persistence diagram. This provides a new invariant for persistent homology over general posets. Finally, we use our homological Galois Connection Theorem to prove several results about the persistence diagram.