Cubical setting for discrete homotopy theory, revisited

2022/02/07 by Carranza, Daniel, Kapulkin, Chris · 5 citations
#05C25 #18N40 #18N45 (secondary) #55U35 (primary) #Algebraic Topology (math.AT) #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.03516

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.

Cited by

• Fast computation of the first discrete homology group
• Biclosed monoidal structures on the categories of digraphs and graphs
• Stability of persistent path homology of path complexes
• Derived mapping spaces of ∞-categories
• On the homology of simplicial and cubical sets with symmetries
• Thomason-Type Model Structures on Simplicial Complexes and Graphs

Related

• Homotopy groups of cubical sets
• A Homotopy Theory for Graphs
• Homotopy n-types of cubical sets and graphs
• Higher discrete homotopy groups of graphs
• Varieties of Cubical Sets