Abstract

In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations.

Highlights

  • In this paper, we apply the methods of the path homology theory of digraphs and quivers first defined in [1,2,3,4,5,6] to the category and homotopy category of vertex colored digraphs and graphs.We construct the collection of path homology theories for vertex colored digraphs

  • The path homology theory is a natural generalization of simplicial homology theory and is defined for any path complex

  • We prove the invariance of the colored-path homology groups relative to the colored homotopy

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Summary

Introduction

We apply the methods of the path homology theory of digraphs and quivers first defined in [1,2,3,4,5,6] to the category and homotopy category of vertex colored digraphs and graphs (see [7,8,9,10,11]). The path homology groups do not have any structure that is similar to the n-skeleton It follows directly from the results below that a vertex coloring of a digraph gives a functorial filtration that agrees with path homology groups. Let us mention here that the application of the colored homology theory to the case of quivers requires significant modifications This fact follows from the generalization of the path homology theory to the category of quivers constructed in [4]. We describe the basic properties of the colored homotopy and colored morphisms and construct a collection of functorial path homology theories for colored digraphs.

The Path Homology Theory for Digraphs
Categories of Colored Digraphs
Path Homology of Colored Digraphs
Spectral Sequence for Path Homology Groups of a Colored Digraph
Path Homology of Colored Graphs
Conclusions
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