Model categories 13 — Find a model category you can prove is not cofibrantly generated.

Definitions and mathematical statement

Exhibit a model category $(\mathcal M,W,\mathrm{Cof},\mathrm{Fib})$ and prove that its given model structure is not cofibrantly generated. In the standard definition, cofibrant generation requires sets $I,J$ with \[ I\text{-inj}=W\cap\mathrm{Fib},\qquad J\text{-inj}=\mathrm{Fib}, \] where inj denotes the right lifting property, together with the smallness conditions needed for the small-object factorizations. The required negative proof must rule out all such generating sets, not merely one proposed choice.

The source proposes a chain-complex model whose weak equivalences are chain homotopy equivalences. In a standard unbounded $R$-module version, cofibrations are degreewise split monomorphisms and fibrations degreewise split epimorphisms; a weak equivalence has an inverse up to chain homotopy. The ring, boundedness and exact variant are not given in the source and must be fixed before proving non-generation. Ring dependence is essential: over a field, splitting makes the homotopy-equivalence and quasi-isomorphism notions coincide, so an unqualified claim that every such chain model is non-generated cannot be used.

The source credits Mike Cole with an early written description and speculates Quillen may have known the structure. The summary reports a later non-generation result but offers ambiguous "and/or" references and no ring hypotheses; those reports are not themselves a proof for the unspecified candidate.

Short English statement

Give a model category with a proof that no sets of generating cofibrations and trivial cofibrations can define its model structure; the proposed chain-homotopy model needs a specified base ring.

Sources

• Mark Hovey, Algebraic Topology Problem List, "Model categories," item 13, surviving mirror https://www-users.cse.umn.edu/~tlawson/hovey/. Recorded access 2026-07-29; extraction html-ordered-list. Mike Cole and Quillen are mentioned without full references.
• Supplied summary: Christensen--Hovey, "Quillen model structures for relative homological algebra" (2001); purported Christensen--Raptis "Six model structures for DG-modules" and "Realizing spectra in ..."; and Raptis's thesis. The summary's author/title assignments and alternatives are incomplete and unverified, not reconstructed into a reliable citation.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11100059$, exact source code catalogue entry; registry open/OPEN-TRIAGE with a solved report in the summary.