[Paper Review] Are all cofibrantly generated model categories combinatorial?

Jir̆ı́ Rosický|arXiv (Cornell University)|May 5, 2009

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper establishes that Vopěnka’s principle is logically equivalent to the statement that every cofibrantly generated model category (in a generalized sense) is Quillen equivalent to a combinatorial model category. Using the trivial model structure on a bounded, non-locally presentable category, the authors prove that the existence of such a Quillen equivalence implies Vopěnka’s principle, thereby showing the set-theoretic necessity of the axiom for Raptis' original result.

ABSTRACT

G. Raptis has recently proved that, assuming Vopěnka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial one. His result remains true for a slightly more general concept of a cofibrantly generated model category. We show that Vopěnka's principle is equivalent to this claim. The set-theoretical status of the original Raptis' result is open.

Motivation & Objective

• To determine whether Vopěnka’s principle is necessary for Raptis’ result that every cofibrantly generated model category is Quillen equivalent to a combinatorial one.
• To investigate the set-theoretic status of Raptis’ theorem by analyzing the relationship between cofibrantly generated and combinatorial model categories.
• To explore whether the equivalence between cofibrantly generated and combinatorial model categories holds without assuming Vopěnka’s principle.
• To establish that the existence of a Quillen equivalence to a combinatorial model category implies Vopěnka’s principle, using a counterexample category.

Proposed method

• Construct a trivial model structure on a bounded, cocomplete category K that is not locally presentable, using a full subcategory A of rigid graphs indexed by ordinals.
• Use the canonical functor E_A: K → Set^{A^op} to analyze density and representability, showing that A is dense but K is not locally presentable.
• Prove that the weak factorization system (Iso, K) is cofibrantly generated in the generalized sense but not in the standard sense due to non-presentable objects.
• Assume a Quillen equivalence between the trivial model category K and a combinatorial model category M, and derive a contradiction via colimit preservation and factorization properties.
• Use the existence of a regular cardinal λ₀ such that the replacement functor R preserves λ₀-filtered colimits in M, leading to a contradiction when id₁ factors through a component of a colimit cocone.
• Apply results from [8] and [7] to show that if K is the full image of a colimit-preserving functor from M, then K must be locally presentable — contradicting the assumption.

Experimental results

Research questions

• RQ1 Is Vopěnka’s principle necessary for every cofibrantly generated model category to be Quillen equivalent to a combinatorial one?
• RQ2 Can a cofibrantly generated model category that is not locally presentable still admit a Quillen equivalence to a combinatorial model category?
• RQ3 Does the generalized definition of cofibrantly generated weak factorization systems (based on cofibrantly closed classes) yield stronger results than the standard definition?
• RQ4 Is the trivial model structure on a bounded but non-locally presentable category cofibrantly generated in the generalized sense?
• RQ5 Does the existence of a combinatorial model structure Quillen equivalent to a trivial model category imply local presentability?

Key findings

• Vopěnka’s principle is logically equivalent to the statement that every cofibrantly generated model category (in the generalized sense) is Quillen equivalent to a combinatorial model category.
• The trivial model structure on a bounded, cocomplete category K that is not locally presentable provides a counterexample where (K, Iso) is cofibrantly generated in the generalized sense but not in the standard sense.
• The existence of a Quillen equivalence between K and a combinatorial model category M implies that K must be locally presentable, contradicting the construction of K under the negation of Vopěnka’s principle.
• The proof shows that if such a Quillen equivalence exists, then the identity morphism on the terminal object 1 would factor through a component of a colimit cocone, contradicting the non-presentability of 1.
• Under the assumption of a proper class of compact cardinals, a trivial model category K has a combinatorial model if and only if it is locally presentable.
• Thus, without Vopěnka’s principle, there exist cofibrantly generated model categories without a Quillen equivalent combinatorial model.