Cofibration category of digraphs for path homology

Carranza, Daniel; Doherty, Brandon; Kapulkin, Chris; Opie, Morgan; Sarazola, Maru; Wong, Liang Ze

doi:10.48550/arxiv.2212.12568

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Mh a Spectral Sequence And Homotopy Relations1

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“…Theorem 4.37 suggests that there should be some homotopy theory on Fsetcat that coincides with the above Ciricis' structure when we apply a functor FC • : Fsetcat −→ FCh ≥0 . The work [4] by Carranza et al seems to support this hypothesis. In that paper, they constructed a cofibration category structure on the category of digraphs, where weak equivalences are exactly the morpshisms inducing isomorphisms on the path homology, namely a part of E 2 .…”

Section: Mh a Spectral Sequence And Homotopy Relationsmentioning

confidence: 84%

Geometric approach to graph magnitude homology

Asao

Izumihara

2021

Homology, Homotopy and Applications

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In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.

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Section: Mh a Spectral Sequence And Homotopy Relationsmentioning

confidence: 84%