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Discrete homotopy hypothesis for n-types

Daniel Carranza, Chris Kapulkin

Abstract

We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical set to a graph, we are also able to give explicit computations of several previously unknown discrete homotopy groups of boundaries of cubes and suspensions of cycles.

Discrete homotopy hypothesis for n-types

Abstract

We show that discrete and classical homotopy theories are equivalent after localizing at n-equivalences for any non-negative integer n. By constructing an explicit homotopy inverse to the graph nerve functor associating an n-fibrant cubical set to a graph, we are also able to give explicit computations of several previously unknown discrete homotopy groups of boundaries of cubes and suspensions of cycles.
Paper Structure (21 sections, 73 theorems, 127 equations, 7 figures, 1 table)

This paper contains 21 sections, 73 theorems, 127 equations, 7 figures, 1 table.

Table of Contents

  1. Organization of the paper.
  2. Acknowledgements.
  3. Preliminaries
  4. Graphs and discrete homotopy theory
  5. Cubical sets
  6. Semicubical sets
  7. Nerves of graphs
  8. Double mapping cylinders
  9. $n$-Skeletal pushouts
  10. Mapping cylinders
  11. Comparison with the cubical double-mapping cylinder
  12. The inverse construction
  13. Explicit description in terms of $F$-sequences.
  14. Main theorem
  15. Computing explicit discrete homotopy groups
  16. ...and 6 more sections

Key Result

Theorem 1

For any non-negative integer $n$, the graph nerve functor $\operatorname{N}_{\infty} \colon \mathsf{Graph} \to \mathsf{cSet}$ is an equivalence after localizing both sides at $n$-equivaelnces.

Figures (7)

  • Figure 1: The graphs $I_3$ and $C_3$, respectively.
  • Figure 2: The double mapping cylinders described in \ref{['ex:boundary-as-cyl']}.
  • Figure 3: The length-3 suspension $\Sigma_{3}{C_5}$ of the 5-cycle.
  • Figure 4: Depictions of the latching maps for $F$.
  • Figure 5: The graphs $F(2, 0)$ and $F(2, 1)$. The blue vertices denote the boundary $\partial F(2, 1)$.
  • ...and 2 more figures

Theorems & Definitions (169)

  • Theorem : cf. \ref{['nerve-equiv-main-thm']}
  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • proposition 8
  • definition 9
  • ...and 159 more