Cofibration category of digraphs for path homology

TL;DR

The paper constructs a cofibration-category framework on the category of directed graphs where weak equivalences are maps inducing isomorphisms in path homology. It introduces cofibrations via a no-outgoing-edges condition together with projecting decompositions, and develops excision to ensure pushouts preserve relative path-homology. The main theorem establishes the full cofibration-category structure, with careful attention to factorization, transfinite composition, and stability properties, while also clarifying interactions with chain-complex models and highlighting non-monoidal behavior. This provides a robust homotopical toolkit for path-homology in digraphs and connects to broader categorical approaches to generalized cohomology theories.

Abstract

We prove that the category of directed graphs and graph maps carries a cofibration category structure in which the weak equivalences are the graph maps inducing isomorphisms on path homology.

Figures (20)

• Figure 1: Depictions of alternating cycles
• Figure 2: Depictions of $C_{m,n}$ cycles
• Figure 3: The graph $X = C_{3,1}$ with induced subgraphs $A$ (in red) and corresponding $X^A$ (circled in blue). Only the first two examples admit projecting decompositions.
• Figure 4: The inclusion of the edge $2-3$ (in red) into the commuting square $X = C_{2,2}$ is a cofibration.
• Figure :

Theorems & Definitions (136)

• Theorem : cf. \ref{['cofib-cat']}
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Example 1.5
• Definition 1.6
• Definition 1.7
• Definition 1.8
• Definition 1.9