Discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to homotopy groups of geometric realizations, giving combinatorial models for arbitrary cubical sets.
The discrete homotopy hypothesis for directed graphs
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
abstract
We develop a homotopy theory of directed graphs based on cubical homotopy groups, also referred to as A-groups or reduced GLMY homotopy groups. Localizing the category of directed graphs at morphisms that induce isomorphisms on these groups yields an $\infty$-category, which we denote by ${\sf DGra}_\infty$. Our main result shows that ${\sf DGra}_\infty$ is equivalent to the $\infty$-category of spaces.
fields
math.AT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.
citing papers explorer
-
Discrete homotopy groups of cubical sets
Discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to homotopy groups of geometric realizations, giving combinatorial models for arbitrary cubical sets.
-
Homotopy theories via the magnitude-path spectral sequence
Defines r-quasi-isomorphisms and r-cofibrations on generalized metric spaces so that each page of the magnitude-path spectral sequence satisfies metric Eilenberg-Steenrod axioms and supports Brown category structures for homotopy colimits, restricting to directed graphs at r=1.