Inserted: 4 jan 2022
Last Updated: 21 apr 2023
Journal: Journal of Algebraic Combinatorics
A cohomology theory for digraphs, called multipath cohomology, has been recently introduced by the authors. Its construction uses the so-called path poset, i.e. the poset of disjoint oriented simple paths in a digraph. The aim of this paper is to explore some combinatorial properties of the path posets, in order to provide new insights and computations. In particular, we develop acyclicity criteria for multipath cohomology, and compute it for oriented linear graphs. To conclude, we interpret multipath cohomology as the cohomology of a certain simplicial complex, and we investigate which spaces may or may not arise this way.