Abstract

The past decade has seen large literatures develop around two novel invariants of directedgraphs: magnitude homology (due to Leinster, Hepworth and Wilerton) and the path homology ofGLMY theory. Though their origins are quite separate, Asao proved in 2022 that in fact thesehomology theories are intimately related. To every directed graph one can associate a certainspectral sequence - the magnitude-path spectral seguence, or MPss - whose page El is exactliymagnitude homology, while path homology lies along a single axis of page E2

This talk has two subjects: the MPsS as a whole, and its page E’, which we call the bigraded pathhomology of a directed graph. l wil explain the construction of the sequence and argue that eachone of its pages deserves to be regarded as a homology theory for digraphs, satisfying a knnethformula and an excision theorem, and with a homotopy-invariance property that grows stronger aswe turn the pages of the seguence. The second half of the talk will focus on bigraded pathhomology, which shares the important homological properties of ordinary path homology but is astrictly finer invariant - capable of distinguishing, for example, the directed n-cycles for all n > 2We wil close with some speculations on the implications of all this for the formal homotopy theoryof the category of directed graphs.