Model categories 5 — The second step: show that the category of algebras over a cofibrant operad admits a model structure,…

Definitions and mathematical statement

Let $\mathcal V$ be the symmetric monoidal model category intended by the preceding, omitted step, with its intended model structure on operads. For an operad $O$, let $\operatorname{Alg}_O(\mathcal V)$ be its algebra category and $U_O$ the forgetful functor to $\mathcal V$.

Hovey's Model categories item 5 asks for three statements:

1. For a cofibrant operad $O$, construct a model structure on $\operatorname{Alg}_O(\mathcal V)$ in which a map is a fibration or weak equivalence exactly when its underlying map under $U_O$ is one.
2. For a weak equivalence $\varphi:O\to P$ of cofibrant operads, show that extension and restriction of scalars \[ \varphi_!:\operatorname{Alg}_O(\mathcal V) \rightleftarrows\operatorname{Alg}_P(\mathcal V):\varphi^* \] form a Quillen equivalence, with extension the left adjoint.
3. Relate an $E_\infty$ operad to a cofibrant approximation $Q\operatorname{Comm}\to\operatorname{Comm}$ of the commutative-ring operad. Such an approximation has cofibrant source and is a weak equivalence in the specified operad model structure.

The source considers the last assertion probably already known. It supplies no ambient category, transfer/admissibility, smallness, monoidal-unit, arity or operadic weak-equivalence hypotheses, so these are not unconditional theorems for arbitrary monoidal model categories.

Definitions of $E_\infty$ may impose aritywise contractibility and symmetric-group cofibrancy without full cofibrancy as an operad. Thus a cofibrant resolution can provide an $E_\infty$ model under suitable conventions, but not every object called $E_\infty$ is automatically cofibrant in every operad model structure. Nor does the second statement alone rectify $E_\infty$ algebras to strict commutative algebras when $\operatorname{Comm}$ is not cofibrant. The registry's solved label must be read with these absent hypotheses resolved.

Short English statement

Transfer model structures to algebras over cofibrant operads, compare them under weakly equivalent cofibrant operads, and interpret $E_\infty$ structures through cofibrant commutative-operad resolutions. The ambient hypotheses are essential.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11100051$, exact source code catalogue entry, attributed to Mark Hovey.
• Hovey's Algebraic Topology Problem List, Model categories item 5: https://www-users.cse.umn.edu/~tlawson/hovey/. Supplied access date: 2026-07-29; extraction: html-ordered-list; 65 top-level items in nine category pages.
• The preceding "first step" and its ambient setup are absent. The registry declares all three goals solved without an exact primary theorem, author attribution or bibliographic citation.