nLab discrete homotopy theory

Idea

Discrete homotopy theory (also known as A-homotopy theory) is an area of mathematics concerned with using techniques of homotopy theory to study combinatorial properties of graphs. It does so by introducing a new combinatorial notion of homotopy between graph maps and subsequently reinterpreting the usual homotopy-theoretic invariants, such as homotopy or homology groups, through the lenses of this new notion.

References

• Hélène Barcelo, Reinhard Laubenbacher: Perspectives on A-homotopy theory and its applications, Discrete Mathematics 298 1-3 (2005) 39–61 [doi:10.1016/j.disc.2004.03.016]

• Rachel Hardeman Morrill: The Lifting Properties of A-Homotopy Theory, Contributions to Discrete Mathematics 19 3 (2024) 36-67 [arXiv:1904.12065, doi:10.55016/ojs/cdm.v19i3.69171]

• Chris Kapulkin, Udit Mavinkurve: The fundamental group(oid) in discrete homotopy theory, Advances in Applied Mathematics 164 (2025) 102838 [arXiv:2303.06029, doi:10.1016/j.aam.2024.102838]