Path homology theory of multigraphs and quivers

Abstract

We construct a new homology theory for the categories of quivers and multigraphs and describe the basic properties of introduced homology groups. We introduce a conception of homotopy in the category of quivers and we prove the homotopy invariance of homology groups.

Morse Theory without Non-Degeneracy

“…If we also assume that Γ has no multiple arrows between the same two vertices, then we can make  into a differential graded algebra where the differential of an arrow from to is the sum of all the paths of length 2 from to , and is extended to paths via the Leibniz rule (cf. [40]). Let 2 is given by the sum of all paths of length 2 in Γ; indeed ( Γ ) is given by the sum of all paths of length r in Γ for any ⩾ 1 and (( Γ ) ) is 0 if is even and ( Γ ) +1 if is odd.…”

Multilayer networks as embodied consciousness interactions. A formal model approach.

“…Some concrete research lines are a) giving a concrete phenomenological and biological account for specific layers configurations, b) finding an algebraic and categorical way to describe the graph contraction among the splitting of rotation multilayers, discussed in section 5; c) study new features arising after going back the multilayers to parallel, non-conscious landscapes; d) finding a mathematical way to measure the interaction (the machinery of Fuzzy Logic may be a good starting point). Other possibilities include the use of topological tools such as sheaves theories [55], homotopy [56] and differential geometry.…”

Path homology of directed hypergraphs

“…Recently, the topological properties of digraphs, hypergraphs, multigraphs, and quivers have been studied using various (co)homologies theories, consult e.g. [3], [4], [6], [16], [15], [14], [10], [13].…”

“…In this paper, we construct several functorial and homotopy invariant homology theories on the category of directed hypergraphs using the path homology theory introduced in [8], [10], [12], [13], and [14].…”