Cubical setting for discrete homotopy theory, revisited

TL;DR

The paper builds a cubical nerve construction $N G$ from graphs, proves it is a Kan complex, and shows the discrete A-homotopy groups satisfy $A_n(G,v)\cong \pi_n(NG,v)$. This yields a proof of Babson–Barcelo–de Longueville–Laubenbacher’s conjecture without the cubical approximation hypothesis and a strong discrete Hurewicz theorem. It additionally develops an abstract homotopy-theoretic framework for graphs, equipping Graph with a fibration-category structure and a lax monoidal/enriched, higher-categorical perspective via the nerve. Collectively, these results bridge discrete homotopy theory on graphs with cubical homotopy theory and higher category theory, enabling transfer of tools and insights across domains.

Abstract

We construct a functor associating a cubical set to a (simple) graph. We show that cubical sets arising in this way are Kan complexes, and that the A-groups of a graph coincide with the homotopy groups of the associated Kan complex. We use this to prove a conjecture of Babson, Barcelo, de Longueville, and Laubenbacher from 2006, and a strong version of the Hurewicz theorem in discrete homotopy theory.

Figures (10)

• Figure 1: An example depiction of a graph.
• Figure 2: The graphs $I_3$ and $C_3$, respectively.
• Figure 5: For a 1-cube $f \colon I_1 \to G$ of $\mathrm{N}_{1}{G}$, the map $l^* \colon \mathrm{N}_{1}G \to \mathrm{N}_{2} G$ sends $f$ to the 1-cube $fl \colon I_2 \to G$ of $\mathrm{N}_{2}{G}$ whereas $r^* \colon \mathrm{N}_{1} G \to \mathrm{N}_{2} G$ sends $f$ to the 1-cube $fr \colon I_2 \to G$ of $\mathrm{N}_{2}{G}$.
• Figure 6: For a 2-cube $g \colon I_1^{\otimes 2} \to G$ of $\mathrm{N}_{1}{G}$, the map $l^* \colon \mathrm{N}_{1}G \to \mathrm{N}_{2} G$ sends $g$ to the 1-cube $gl^{\otimes 2} \colon I_2^{\otimes 2} \to G$ of $\mathrm{N}_{2}{G}$ whereas $r^* \colon \mathrm{N}_{1} G \to \mathrm{N}_{2} G$ sends $g$ to the 1-cube $gr^{\otimes 2} \colon I_2^{\otimes 2} \to G$ of $\mathrm{N}_{2}{G}$.
• Figure 7: Vertices of $I_6^{\otimes 2}$ labelled by their image under $d \colon I_6^{\otimes 2} \to I_\infty$.

Theorems & Definitions (159)

• Theorem : cf. \ref{['thm:main', 'thm:a_eq_cset_pi']}
• Theorem : Conjectured in babson-barcelo-longueville-laubenbacher; cf. \ref{['thm:conj-bbdll']}
• Theorem : Discrete Hurewicz Theorem; cf. \ref{['graph-hurewicz']}
• definition 1
• proposition 2
• proof
• remark 3
• definition 4
• remark 5
• proposition 6