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If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-enriched cofibrantly generated model category can be upgraded to an enriched functor. Unfortunately, many interesting $\mathcal{V}$ fail to satisfy this hypothesis. In particular, most monoidal models of spectra will not have all objects cofibrant or the weaker conditions that appear in Theorem 13.5.2. Are there even weaker conditions which guarantee such an enrichment?

I am interested particularly in whether $\mathrm{Spec}$-enriched presheaf categories $\mathrm{Fun}^{\mathrm{Spec}}(C,\mathrm{Spec})$ have an enriched cofibrant replacement with respect to the projective model structure where we model $\mathrm{Spec}$ by $S$-modules. The existence of the projective model structure was proven in Theorem 7.2 of Equivalences of monoidal model categories.

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    $\begingroup$ As you allude to, 13.5.2 doesn't actually require all objects to be cofibrant. What it requires is that $V \otimes (-)$ be left Quillen for each $V \in \mathcal V$. Do you have a counterexample to this property holding in your case? I might think it holds, at least if the "spaces" going into the construction of the model structure are actually simplicial sets -- I would try to prove this by first treating the case where $V$ is a suspension spectrum, and then generalizing by passing to some filtered colimit. $\endgroup$
    – Tim Campion ♦
    Commented Aug 30, 2023 at 13:59
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    $\begingroup$ @TimCampion The reason I think this condition is especially severe is because if you are trying to demonstrate it for $\mathcal{V}$ enriched over itself and $1 \in \mathcal{V}$ is cofibrant, then it implies any $V$ is cofibrant. Perhaps it could be true in the positive model structure on symmetric spectra of simplicial sets? $\endgroup$
    – Connor Malin
    Commented Aug 30, 2023 at 18:22

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I really like this question. Let $\mathcal{V}$ be a monoidal model category of spectra. A $\mathcal{V}$-enriched cofibrant replacement functor is the same thing as a lax monoidal cofibrant replacement functor. Dmitri Pavlov has figured out conditions for such functors to exist, in Section 8 of his preprint Coadmissibility of colored cooperads in monoidal model categories. Example 8.7 shows that his method applies to the category of symmetric spectra, and it's also easy to apply it to other flavors of spectra built on simplicial sets, or to symmetric/orthogonal spectra built on compactly generated spaces. Pavlov's Prop 8.3 explains when having an enriched cofibrant replacement passes to diagram categories, and his 8.5 is about right-induced model structures. Pavlov uses the term oplax monoidal. This has to do with the direction of the morphism $QA \otimes QB \to Q(A\otimes B)$, which you can construct via lifting. Lastly, I'll note that strict cofibrant replacement functors are extremely rare.

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  • $\begingroup$ This is very useful; how precisely does the fact that both $QA \otimes QB$ and $Q(A \otimes B)$ are cofibrant translate to producing natural morphisms either way between them? $\endgroup$
    – Connor Malin
    Commented Jan 15 at 20:40