Result 312, Topology

The Grothendieck homotopy hypothesis

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.

Lean formalization Proof

The bigger picture

Why it matters

The central idea is that algebra can capture the homotopy theory of spaces, not just selected measurements of their shape. The manuscript reports this correspondence for an entire specified class of structures called higher groupoids.

What changes?

A weak globular infinity-groupoid records points, paths between points, and higher-dimensional relations between paths, with operations reversible up to further relations. A coherator specifies the operations and their coherence rules. The manuscript claims that these groupoids recover the homotopy theory of spaces for every Grothendieck coherator in the Ara–Henry convention, not merely a particular choice. It also claims Henry's pushout conjecture: elementary expansions, basic enlargement steps for cellular infinity-groupoids, preserve connected components and all homotopy groups.

What does that help mathematicians do?

The expansion result gives researchers a specific construction step that leaves these homotopy measurements unchanged. Connected components distinguish separate pieces; homotopy groups detect higher-dimensional features through maps from spheres. Preserving both means such expansions cannot create or remove features detected by these invariants. The broader correspondence says that choosing any coherator within the stated convention still yields an algebraic framework that recovers the homotopy theory of spaces.

Are there practical applications?

The immediate value is foundational: it supports studying topological spaces through algebraic descriptions of paths and their higher relations. The expansion claim supplies a way to enlarge cellular descriptions without changing their homotopy groups or components. The supplied material describes no computational method or practical deployment; its contribution is to the mathematical framework for comparing algebra and topology.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Grothendieck homotopy hypothesis via elementary expansions

September 24, 2026 26 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/312.md.

The Grothendieck homotopy hypothesis

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)(n+1)-disk along its source nn-face induces a weak equivalence.

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

Comparator links

Result Comparator statement
Elementary-expansion weak equivalence GrothendieckElementaryExpansion.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.