Abstract:In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentialsbetween magnitude homologies of a digraph G, which make them chain complexes. Then we show that its homology MH ℓ 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 MH k k (G) is isomorphic to the redu… Show more
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