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Consider the global projective model category of simplicial presheaves on some category (the category of smooth manifolds is particularly interesting to me).

In Section 9.1 of Dugger's paper “Universal homotopy theories” one can find a sufficient condition for a simplicial presheaf to be cofibrant, together with descriptions of two different cofibrant replacement functors.

Is there a nontrivial necessary condition for a simplicial presheaf to be cofibrant in the global projective model structure that is easier to check than the lifting property? Such a condition could be used to easily establish noncofibrancy of some simplicial presheaves.

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up vote 4 down vote accepted

Let $\mathscr C$ be a small category. Necessary and sufficient conditions for a presheaf $F$ to be cofibrant in the global projective model structure on $[\mathscr C^\mathrm{op}, [\Delta^\mathrm{op}, \mathbf{Set}]]$ are that:

(1) Each $F(-)(n) \colon \mathscr C^\mathrm{op} \to \mathbf{Set}$ is projective (i.e., a coproduct of retracts of representables; if $\mathscr C$ is Cauchy-complete, then equivalently a coproduct of representables).

(2) $F \colon \mathscr C^\mathrm{op} \to [\Delta^\mathrm{op}, \mathbf{Set}]$ factors through the subcategory $[\Delta^\mathrm{op}, \mathbf{Set}]_{\mathrm{nd}}$ of $[\Delta^\mathrm{op}, \mathbf{Set}]$ whose objects are simplicial sets and whose maps are simplicial maps which sends non-degenerate simplices to non-degenerate simplices.

On the one hand, it's easy to show by induction that any cofibrant object satisfies (1) and (2). Conversely, condition (2) means that each $F(-)(n)$ breaks up as a coproduct $G_n(-) + H_n(-)$ of non-degenerate and degenerate simplices, and the object $G_n(-)$ of non-degenerate simplices is projective since $F(-)(n)$ is. This means that any object satisfying (1) and (2) can be built up dimension by dimension using the generating cofibrations: first construct the $0$-skeleton using pushouts of maps

$\partial \Delta_0 \cdot \mathscr C(-, W) \to \Delta_0 \cdot \mathscr C(-, W)$ (for $W \in \mathscr C$)

one for each representable summand in the projective object $F(-)(0)$; then construct the $1$-skeleton by using pushouts of maps

$\partial \Delta_1 \cdot \mathscr C(-, W) \to \Delta_1 \cdot \mathscr C(-, W)$ (for $W \in \mathscr C$)

one for each representable summand in the projective object $G_1(-)$ of non-degenerate $1$-simplices; and so on.

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Wonderful, just what I looked for! Is this construction described in some reference or should I simply cite this answer? –  Dmitri Pavlov Apr 12 '13 at 5:51
    
It's not written up explicitly anywhere I know, but it's pretty much in the paper of Dugger you cite. –  Richard Garner Apr 15 '13 at 23:34
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The only necessary condition I can think of is that it has to be levelwise cofibrant. But in my work, this is usually a good enough test for establishing noncofibrancy of things. I've also come to believe that there's no nice characterization for when a diagram is projectively cofibrant. For instance, the dual problem (regarding fibrant objects in the injective model structure) was asked here and they didn't really come up with a good answer. In fact, my advisor Mark Hovey commented that a good answer is basically impossible in his opinion.

Sadly, it's not all that hard for a diagram to be levelwise cofibrant without being projectively cofibrant. This phenomenon is discussed in depth here and here (well, this second one isn't really about diagram categories, but it's the motivation for the terminology). I know this probably isn't the answer you wanted, but this question has been open for almost a year and it seemed time to at least have some answer on the table. Here are some more ideas...

If you need a sharper necessary condition, perhaps you can look at the details of how the projective model structure comes about. Hirschhorn is a good source, or you can look at this MO post where I've shortened and distilled his work. One thing this does is give you an exact characterization of the projective cofibrations, and hence of projectively cofibrant objects. But I've never found this useful in practice, since it basically boils down to the lifting you didn't want to check. Anyway, the characterization is as the weakly saturated class of morphisms generated by $Id_{C(X,-)}\cdot i$ where $i$ runs through all generating cofibrations and $\cdot$ is tensoring over Set (i.e. you take $C(X,-)$ many copies of $i$).

For simplicial presheaves you have a little bit more, because you have a Quillen adjunction to the projective model structure on $[C^{op},Ch_\bullet^+]$ where $Ch_\bullet^+$ has the projective model structure. You can use this to test whether or not a diagram is cofibrant based on what happens to it in this other category. This can't be used to establish cofibrancy of the diagram because the adjunction goes the wrong way, but the fact that the adjunction preserves trivial fibrations can still be used to test whether cofibrancy fails.

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Apparently, the fact that the target is such a special category (simplicial sets) allows us to write very concrete necessary and sufficient conditions, as Richard Garner shows in his answer. –  Dmitri Pavlov Apr 12 '13 at 6:16
    
Yeah, thanks for drawing my attention to it. Glad your problem finally got resolved all this time later –  David White Apr 12 '13 at 18:52
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