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The category of enriched functors from finite based CW complexes to based topological spaces has a projective model structure. The fibrations are the objectwise Serre fibrations and the weak equivalences are the objectwise weak homotopy equivalences. The cofibrations are defined by the left lifting property. In particular every cofibration is an objectwise Hurewicz cofibration of based spaces, but probably not an objectwise Serre cofibration.

Is this model structure cellular in the sense of Definition 12.1.1 of Hirschhorn's Model Categories and Their Localisations? It is known to be cofibrantly generated.

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Hi. Welcome to MathOverflow. I posted an answer, but realized it was saying something stupid and now have pulled it back. I will repost if I can sort things out in my head. For now, I recommend checking out this question mathoverflow.net/questions/76160/… and the answers there, which delve into Hirschhorn's Theorem 11.6.1 in detail, and might be useful in transferring cellularity in the way you need. Do you have a good sense of the generating (trivial) cofibrations in your case? –  David White Jun 20 '13 at 17:23
Is it true that if we take all functors rather than just enriched functors that $Fun(CW_\ast,Top_\ast)$ is cellular, e.g. by Proposition 12.1.5? So the problem I guess is that you're looking at a subcategory (now the enriched functors) and a priori there might be enriched functors which are cofibrant here but are not cofibrant in $Fun$ when you forget the enrichment. Still, because this is a transferred model structure you should have (1) and (2) of Hirschhorn for free, leaving us only to wonder if objectwise Hurewicz cofibrations are effective monomorphisms. Am I missing something? –  David White Jun 20 '13 at 18:17
Thanks for the welcome! I agree that the category of all functors should be cellular. But when working in the enriched setting, I can't seem to show the compactness conditions. The problem is that evaluating a cell complex (in the category of enriched functors) doesn't seem to give a cell complex of spaces. Instead it is just a sequence of $h$-cofibrations. –  David Barnes Jun 21 '13 at 8:12
In the usual situation of diagram categories the projective generating (triv) cofibrations are just products of the old ones. Is that the case here? If so, shouldn't the domains just be products of spheres and hence compact as topological spaces? Then you can apply 2.4.2 in Hovey's book to say they're compact relative to closed $T_1$ inclusions. Does that help at all? I feel like a sequence of $h$-cofibrations should be a closed $T_1$ inclusion. If this doesn't help, maybe you could get away with just working in Jeff Smith's $\Delta$-generated spaces, which is a combinatorial model of spaces. –  David White Jun 21 '13 at 15:57
The domains of the generating trivial cofibrations have form $CW_*(X,-) \wedge S^n_+$ where $CW_*(X,-)$ is an enriched functor from $CW_*$ to based spaces whose value at $Y$ is the based space of based maps from $X$ to $Y$ (with the compact-open topology). This morning I found Proposition A.8. of Hovey's "Spectra and symmetric spectra in general model categories". This should be applicable to general topological diagram categories, but only when the projective cofibrations are objectwise Serre cofibrations of spaces. Which is not true in my case. –  David Barnes Jun 21 '13 at 16:35
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