[Paper Review] Theories homotopiques de Quillen combinatoires et derivateurs de Grothendieck

Olivier Renaudin|ArXiv.org|Mar 14, 2006

Homotopy and Cohomology in Algebraic Topology7 references6 citations

TL;DR

This paper constructs a pseudo-localization of the 2-category of combinatorial Quillen model categories relative to Quillen equivalences, proving it embeds fully faithfully into the 2-category of Grothendieck derivators with right and left adjunctions. Using Dugger's presentation results and Cisinski's derivator machinery, the authors establish a local equivalence between the homotopy 2-category of combinatorial Quillen theories and the 2-category of right/left adjoint derivators, showing that derivators capture the essential homotopical structure of combinatorial model categories.

ABSTRACT

We construct a pseudo-localization of the 2-category of combinatorial Quillen model categories with respect to Quillen equivalences, and then verify that it embeds in a 2-category of Grothendieck derivators.

Motivation & Objective

• To compare the 2-category of combinatorial Quillen model categories with the 2-category of Grothendieck derivators.
• To construct a pseudo-localization of the 2-category of combinatorial Quillen model categories relative to Quillen equivalences.
• To prove that this pseudo-localization fully embeds into the 2-category of right and left adjoint derivators.
• To show that the resulting 2-category of homotopy theories captures the essential homotopical invariants of combinatorial model categories.
• To extend the comparison to pointed, stable, simplicial, and spectral model categories using Dugger's results.

Proposed method

• Constructs a pseudo-localization of the 2-category of combinatorial Quillen model categories using pseudo-functors that send Quillen equivalences to equivalences.
• Introduces cylinder and path objects in the 2-category of Quillen model categories to define Quillen homotopies as weak equivalences on cofibrant objects.
• Applies Dugger’s presentation theorem to show every combinatorial model category is Quillen equivalent to a presentable one.
• Uses Cisinski’s construction of a pseudo-functor from Quillen model categories to derivators to induce a map from the pseudo-localization to the 2-category of derivators with adjunctions.
• Proves the induced pseudo-functor is a local equivalence using Cisinski’s representability theorem for derivators.
• Introduces the notion of derivators of small presentation to characterize the essential image of the construction.

Experimental results

Research questions

• RQ1 Can the 2-category of combinatorial Quillen model categories be localized at Quillen equivalences to form a well-behaved 2-category of homotopy theories?
• RQ2 Is there a fully faithful embedding of this localized 2-category into the 2-category of Grothendieck derivators with adjunctions?
• RQ3 Do derivators capture the homotopical invariants of combinatorial model categories up to equivalence?
• RQ4 How do cylinder and path objects in the 2-category of model categories ensure that homotopies are sent to isomorphisms under localization?
• RQ5 What is the essential image of the localization map in terms of derivator-theoretic properties?

Key findings

• The 2-category of combinatorial Quillen homotopy theories, denoted $\mathfrak{THQ}^c$, is constructed as a pseudo-localization of the 2-category of combinatorial Quillen model categories at Quillen equivalences.
• The pseudo-localization $\Gamma: \mathfrak{ModQ}^c \to \mathfrak{THQ}^c$ satisfies a universal property: any pseudo-functor sending Quillen equivalences to equivalences factors uniquely through $\Gamma$.
• The construction ensures that Quillen homotopies (weak equivalences on cofibrant objects) are sent to isomorphisms in $\mathfrak{THQ}^c$, reflecting homotopical invariance.
• There exists a local equivalence $\mathfrak{THQ}^c \to \mathfrak{Der}_{ad}$, where $\mathfrak{Der}_{ad}$ is the 2-category of derivators with right and left adjunctions.
• The essential image of this equivalence consists of derivators of small presentation, characterizing the derivators that arise from combinatorial model categories.
• The results extend to pointed, stable, simplicial, and spectral model categories via Dugger’s observations, showing the framework is robust across standard homotopical contexts.