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Homotopy $n$-types of cubical sets and graphs

Chris Kapulkin, Udit Mavinkurve

TL;DR

This work develops a robust framework for homotopy $n$-types in discrete settings by transporting two parallel model-structure constructions from simplicial sets to cubical sets and then applying them to graphs via the graph nerve. It shows that a right-transferred model structure along $\mathrm{cosk}_{n+1}$ yields a cubical $n$-type model whose fibrant objects align with vanishing higher homotopy, and establishes a Quillen equivalence with the canonical Cisinski $n$-type model. The paper then builds a fibration category on graphs with weak equivalences given by discrete $n$-equivalences and proves that for $n=1$ this recovers the classical groupoid model, via the fundamental groupoid functor. Overall, it offers a bridge between classical homotopy theory and discrete graph-homotopy, enabling transfer of techniques like Mayer–Vietoris and Blakers–Massey to combinatorial contexts.

Abstract

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete $n$-equivalences.

Homotopy $n$-types of cubical sets and graphs

TL;DR

This work develops a robust framework for homotopy -types in discrete settings by transporting two parallel model-structure constructions from simplicial sets to cubical sets and then applying them to graphs via the graph nerve. It shows that a right-transferred model structure along yields a cubical -type model whose fibrant objects align with vanishing higher homotopy, and establishes a Quillen equivalence with the canonical Cisinski -type model. The paper then builds a fibration category on graphs with weak equivalences given by discrete -equivalences and proves that for this recovers the classical groupoid model, via the fundamental groupoid functor. Overall, it offers a bridge between classical homotopy theory and discrete graph-homotopy, enabling transfer of techniques like Mayer–Vietoris and Blakers–Massey to combinatorial contexts.

Abstract

We give a new construction of the model structure on the category of simplicial sets for homotopy -types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an analogous model structure can be constructed on the category of cubical sets, and use it to equip the category of (simple) graphs with a fibration category structure whose weak equivalences are discrete -equivalences.
Paper Structure (9 sections, 48 theorems, 57 equations)

This paper contains 9 sections, 48 theorems, 57 equations.

Table of Contents

  1. Homotopy n-types of simplicial sets
  2. Homotopy $n$-types of cubical sets
  3. Background on cubical sets
  4. The Cisinski model category structure for $n$-types of cubical sets
  5. The transferred model category structure for $n$-types of cubical sets
  6. Application to discrete homotopy theory
  7. Background on discrete homotopy theory
  8. Homotopy $n$-types of graphs
  9. The fundamental groupoid in discrete homotopy theory

Key Result

theorem 1

The category $\mathsf{sSet}$ of simplicial sets admits a cofibrantly generated model category structure, where: Moreover, a simplicial map $f \colon X \to Y$ between $n$-fibrant simplicial sets $X$ and $Y$ is an $n$-fibration if and only if it is a naive $n$-fibration. We refer to this model category structure as the Cisinski model category structure for $n$-types of simplicial sets, and denote i

Theorems & Definitions (107)

  • theorem 1: cisinski:presheaves
  • definition 3: cf. hess-kedziorek-riehl-shipley
  • theorem 4: cf. Elvira-Donazar_Hernandez-Paricio_1995
  • lemma 5: Elvira-Donazar_Hernandez-Paricio_1995
  • proof
  • proposition 6: Elvira-Donazar_Hernandez-Paricio_1995
  • proof
  • lemma 7
  • proof
  • proposition 8
  • ...and 97 more