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Topology · Result 312

The Grothendieck homotopy hypothesis

1 paperLean

Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.

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Papers 1

First page of The Grothendieck homotopy hypothesis via elementary expansions

The Grothendieck homotopy hypothesis via elementary expansions

OpenAI · September 24, 2026

We prove the Grothendieck homotopy hypothesis for every Grothendieck coherator in the Ara–Henry convention: its weak globular infinity-groupoids recover the homotopy theory of spaces. We also resolve Henry's pushout conjecture, showing that elementary expansions preserve components and all homotopy groups of cellular infinity-groupoids.

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@misc{OAI:The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026,
  author = {{OpenAI}},
  title = {{The Grothendieck homotopy hypothesis via elementary expansions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026/paper.pdf}{OAI:The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The Grothendieck homotopy hypothesis asks whether algebraic \(\infty\)-groupoids recover the homotopy theory of spaces. The formalized result is the elementary-expansion theorem used in this approach: for every Grothendieck coherator in the Ara–Henry convention and every boundary-cellular model, attaching an \((n+1)\)-disk along its source \(n\)-face induces a weak equivalence.

The later semi-model structure and full comparison with the homotopy theory of spaces are not included.

Comparator links

ResultComparator statement
Elementary-expansion weak equivalenceGrothendieckElementaryExpansion.lean

Scope note on GitHub →