Results on Grothendieck infinity-groupoids and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Models by generators — Proposition 3.2. A model of a coherator is a globular set with a choice of each generating operation subject only to its two face equations; morphisms are the globular maps preserving the generating operations.
• Limits, colimits and free models — Theorem 4.3. The models form a complete and cocomplete category with limits and filtered colimits computed on cells and a free model functor; free models on pasting schemes are representable, and free models on finite globular sets are finitely presentable.
• The small object argument — Proposition 5.1. For a set of morphisms with finitely presentable domains, every morphism factors as a countable cellular extension followed by a morphism with the right lifting property.
• Division — Lemma 2.1. Multiplication by a fixed cell at any level below the top induces bijections between homotopy sets with prescribed boundaries.
• The exact-boundary test — Proposition 3.1. A morphism of models is a weak equivalence exactly when every cell of the target with a boundary coming from the domain is homotopic to the image of a cell with exactly that boundary.
• Lifting tests — Lemma 4.2. The morphisms with the right lifting property against all boundary inclusions are exactly the weak equivalences with the right lifting property against all source inclusions.
• The cutoff reflector — Lemma 1.1. The models with unique fillers above dimension q form a reflective subcategory, and the reflection keeps all cells through dimension q.
• Free shapes — Proposition 2.2. Free models on pasting schemes, and their coskeletal reflections, are contractible.
• Upper splitting — Lemma 3.2. Under the descending induction hypothesis, an upper cellular weak equivalence with cellular source has a retraction, and its pushouts along maps to cellular models are weak equivalences.
• Trees are contractible — Proposition 1.1. Free models on tree diagrams at depth n are contractible, using only expansions of index n + 1.
• Whiskered shapes — Proposition 2.1. Free models on pasting schemes with a whisker attached to every n-cell are contractible.
• The partial path object — Proposition 3.1. A functorial partial path object whose two endpoints lift all boundary inclusions and whose joint endpoint map lifts the source inclusion in dimension n.
• Henry's pushout conjecture — Theorem 0.1. For every coherator, every elementary expansion of a cellular model, along an arbitrary cell, is a weak equivalence.
• The canonical semi-model structure — Theorem 2.1. The models carry a cofibrantly generated left semi-model structure with boundary inclusions as generating cofibrations, source inclusions as generating trivial cofibrations, and every object fibrant.
• The homotopy hypothesis (assuming Henry's comparison theorem) — Corollary 3.1. The semi-model category of models of any coherator is Quillen equivalent to topological spaces.

Reading order

Six lessons, to be read in order. The first two set up globular theories, their models and the homotopy calculus. The third to fifth prove the main theorem by descending induction at a fixed cutoff: free shapes, trees and whiskers are contractible, partial cylinders give a path object, and the inductive step follows by a retract argument. The sixth derives the canonical semi-model structure. One theorem is used without proof in the programme: Henry's comparison theorem for cylinder coherators, which gives the Quillen equivalence with spaces.