Homotopy by degrees, and the magnitude-path spectral sequence

The past decade has seen a proliferation of homology theories for graphs. In particular, large literatures have grown up aroundmagnitude homology (due to Hepworth and Willerton) and path homology (Grigor’yan, Lin, Muranov and Yau). Though their origins are quite separate, Asao proved in 2022 that in fact these homology theories are intimately related. To every directed graph one can associate a certain spectral sequence—the magnitude-path spectral sequence, or MPSS—whose page E^1 is exactly magnitude homology, while path homology lies along a single axis of page E^2. In this talk, based on joint work with Richard Hepworth, I will explain the construction of the sequence and argue that each one of its pages deserves to be regarded as a homology theory for directed graphs, satisfying a Künneth theorem and an excision theorem, and with a homotopy-invariance property that grows stronger as we turn the pages of the sequence. The 'nested’ family of homotopy categories associated to the pages is not yet well understood. But I will describe a new cofibration category structure on the category of directed graphs, associated to page E^2 of the MPSS.