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](https://arxiv.org/abs/math/0007070)”
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.



Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves
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.


