Grothendieck infinity-groupoids and the homotopy hypothesis

Henry's pushout conjecture for every Grothendieck coherator, after OpenAI's 2026 proof: freely adjoining to a cellular infinity-groupoid a cell parallel to a given one, and a cell joining the two, is a weak equivalence. Globular theories and their models; the homotopy calculus; coskeletal cutoffs, spines and free shapes; trees and whiskers; partial cylinders and the main theorem; the canonical semi-model structure and the Quillen equivalence with spaces.

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Globular theories and their models
2. The homotopy calculus
3. Cutoffs, spines and free shapes
4. Trees and whiskers
5. Partial cylinders and the main theorem
6. The semi-model structure and the homotopy hypothesis

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• The functor of points and representability · Sheaves and schemes
• Kan extensions and the derivator axioms (Derivators)
• Presheaves, the Yoneda lemma and left Kan extensions (courses on schemes and on derivators); Grothendieck's homotopy programme (course AG-AS, Test categories and Grothendieck's homotopy programme) for context.