A Model of Directed Graph Cofiber

Zachary Mcguirk,Byungdo Park

doi:10.3390/axioms11010032

Zachary Mcguirk, Byungdo Park

Open Access

https://doi.org/10.3390/axioms11010032

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Abstract

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective.

Highlights

In 2014, Grigor’yan, Lin, Muranov, and Yau [13] considered digraph homotopy as a notion of a discrete analog of homotopy in topology and proved that their path-space homology is invariant under this version of homotopy
The path-space homology satisfies all discrete analogs of Eilenberg–Steenrod axioms, but these axioms are not complete, and we note that the completeness of Eilenberg–Steenrod axioms has a topological nature in it
Another example is a failure of homotopy extension property for directed graphs [16], which reveals that the Brown representability theorem for not-necessarily-finite CW complexes is likely to be a special feature of CW complexes rather than a consequence of generalities in category theory

Summary

Introduction

In 2014, Grigor’yan, Lin, Muranov, and Yau [13] considered digraph homotopy as a notion of a discrete analog of homotopy in topology and proved that their path-space homology is invariant under this version of homotopy. The path-space homology satisfies all discrete analogs of Eilenberg–Steenrod axioms, but these axioms are not complete (see [6]), and we note that the completeness of Eilenberg–Steenrod axioms has a topological nature in it Another example is a failure of homotopy extension property for directed graphs [16], which reveals that the Brown representability theorem for not-necessarily-finite CW complexes is likely to be a special feature of CW complexes rather than a consequence of generalities in category theory.

The Category of Directed Graphs

Operations in Directed Graphs

Homotopy for Digraphs

Main Construction

Discussion