This talk introduces Möbius homology, a new homological invariant for modules over finite posets that categorifies the classical Möbius inversion formula. Given a module over a poset (a functor from a poset to an abelian category), we construct its Möbius homology by localizing a simplicial cosheaf over the order complex of the poset. Our main result shows that the Euler characteristic of Möbius homology recovers the Möbius inversion, providing a topological lift of this fundamental combinatorial operation.

The primary motivation for developing this theory comes from persistent homology. We demonstrate how Möbius homology provides a natural categorification of persistence diagrams for modules over arbitrary finite posets, extending beyond the classical totally ordered case.