Geodesic
Homology of transitive digraphs and discrete spaces
Sbornik. Mathematics, Tome 214 (2023) no. 8, pp. 1121-1139

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We prove that for transitive digraphs path homology, and therefore also Alexandroff homology, coincides with singular cubic homology. Also, discrete topological spaces are defined that are natural analogues of standard topological cubes. Using them, the singular cubic homology of discrete topological spaces is defined, and it is proved that these homology groups coincide with the Alexandroff homology groups. Bibliography: 24 titles.
Keywords: transitive digraphs, Alexandroff homology, path homology, discrete topological spaces, digraph homology, homology of discrete topological spaces, singular cubic homology. 
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Yu. V. Muranov; R. Jimenez. Homology of transitive digraphs and discrete spaces. Sbornik. Mathematics, Tome 214 (2023) no. 8, pp. 1121-1139. http://geodesic.mathdoc.fr/item/SM_2023_214_8_a4/