{"paper":{"title":"Magnitude homology and Path homology","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Yasuhiko Asao","submitted_at":"2022-01-20T08:25:13Z","abstract_excerpt":"In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\\mathrm MH}^{\\ell}_k(G) \\longrightarrow {\\mathrm MH}^{\\ell-1}_{k-1}(G)$ between magnitude homologies of a digraph $G$, which make them chain complexes. Then we show that its homology ${\\mathcal MH}^{\\ell}_k(G)$ is non-trivial and homotopy invariant in the context of `homotopy theory of digraphs' developed by Grigor'yan--Muranov--S.-T. Yau et al (G-M-Ys in the following). It is remarkable that the diagonal part of our homology ${\\mathcal MH}^{k}_k(G)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.08047","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.08047/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}