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

Question

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.

Answer 1

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.