[Paper Review] Grothendieck's Homotopy Hypothesis

Ilias Amrani|arXiv (Cornell University)|Dec 6, 2011

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper establishes a cofibrantly generated, left proper, and cellular model structure on simplicial topological categories (sCat_Top), proving that the category of ∞-groupoids—topological categories where all morphisms are homotopy equivalences—is Quillen equivalent to simplicial sets. This equivalence confirms Grothendieck’s homotopy hypothesis, asserting that ∞-groupoids model all homotopy types.

ABSTRACT

We construct a "diagonal" cofibrantly generated model structre on the category of simplicial objects in the category of topological categories sCat_{Top}, which is the category of diagrams [Δ^{op}, Cat_{Top}]. Moreover, we prove that the diagonal model structures is left proper and cellular. We also prove that the category of \infty-groupoids (the full subcategory of topological categories) has a cofibrantly generated model structure and is Quillen equivalent to the model category of simplicial sets, which proves the Grothendieck's homotopy hypothesis.

Motivation & Objective

• To construct a cofibrantly generated, left proper, and cellular model structure on the category of simplicial topological categories, sCat_Top.
• To define and characterize the diagonal model structure such that weak equivalences detect homotopy types via the coherent nerve and realization of ∞-groupoids.
• To prove that the full subcategory of ∞-groupoids in Cat_Top admits a cofibrantly generated model structure with Dwyer-Kan equivalences as weak equivalences.
• To establish a Quillen equivalence between the category of ∞-groupoids and the category of simplicial sets, thereby proving Grothendieck’s homotopy hypothesis.
• To extend the framework to n-groupoids and show that n-types are equivalent to n-groupoids in the homotopy category.

Proposed method

• Construct a diagonal model structure on sCat_Top by lifting weak equivalences and fibrations via the coherent nerve and realization functors.
• Use the left Quillen functor $k^!$ to relate the Kan and quasi-category model structures on simplicial sets, ensuring homotopy types are preserved.
• Prove left properness and cellularity of the diagonal model structure using lifting properties and small object arguments.
• Define the ∞-groupoid associated to a topological category as the Dwyer-Kan localization of its core, and show that the coherent nerve of this localization recovers the homotopy type.
• Establish a Quillen adjunction between ∞-groupoids and simplicial sets via the functors $\Theta: \mathbf{sSet} \to \mathbf{Cat}_{\mathbf{Top}}$ and $\Psi: \mathbf{Cat}_{\mathbf{Top}} \to \mathbf{sSet}$, and prove it is a Quillen equivalence.
• Use the fact that $k^! \widetilde{N}_\bullet \mathrm{sing} \mathbf{C} \to \widetilde{N}_\bullet \mathrm{sing} \mathbf{C}$ is a trivial fibration to show the adjunction induces an equivalence on homotopy categories.

Experimental results

Research questions

• RQ1 Can a cofibrantly generated, left proper, and cellular model structure be defined on the category of simplicial topological categories?
• RQ2 Is the diagonal model structure on sCat_Top Quillen equivalent to the standard model structure on simplicial sets?
• RQ3 Does the category of ∞-groupoids in Cat_Top admit a model structure where weak equivalences are Dwyer-Kan equivalences?
• RQ4 Is the adjunction between ∞-groupoids and simplicial sets a Quillen equivalence, thereby proving Grothendieck’s homotopy hypothesis?
• RQ5 Can the homotopy theory of n-types be recovered from the homotopy theory of n-groupoids?

Key findings

• A cofibrantly generated, left proper, and cellular model structure exists on the category of simplicial topological categories (sCat_Top), called the diagonal model structure.
• The diagonal model structure satisfies the condition that a map $\mathbf{A}_\bullet \to \mathbf{B}_\bullet$ is a weak equivalence if and only if $\mathrm{diag}~{}\widetilde{N}_\bullet \mathbf{A}_\bullet' \to \mathrm{diag}~{}\widetilde{N}_\bullet \mathbf{B}_\bullet'$ is a weak equivalence in simplicial sets.
• The category of ∞-groupoids (topological categories where all morphisms are homotopy equivalences) admits a cofibrantly generated model structure with Dwyer-Kan equivalences as weak equivalences.
• The adjunction between ∞-groupoids and simplicial sets, induced by $\Theta$ and $\Psi$, is a Quillen equivalence, proving Grothendieck’s homotopy hypothesis.
• The homotopy category of n-types is equivalent to the homotopy category of n-groupoids, generalizing the homotopy hypothesis to n-groupoids.
• The diagonal model structure on sCat_Top restricts to a Quillen equivalent model structure on the category of simplicial ∞-groupoids, which is also left proper and cellular.