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Homotopy theories via the magnitude-path spectral sequence
Muriel Livernet, Emily Roff, Sarah Whitehouse
We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS…
Bigraded path homology and the magnitude-path spectral sequence
Richard Hepworth, Emily Roff
Two important invariants of directed graphs, namely magnitude homology and path homology, have recently been shown to be intimately connected: there is a 'magnitude-path spectral s…
The reachability homology of a directed graph
Richard Hepworth, Emily Roff
The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and…
Iterated magnitude homology
Emily Roff
Magnitude homology is an invariant of enriched categories which generalizes ordinary categorical homology -- the homology of the classifying space of a small category. The classify…