Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be its image via the Yoneda embedding. Is it true that $y(C_\bullet)$ is a cofibrant object in the local projective model structure? I am particularly interested when $\mathcal{C}=Diff$, and the localization is along the Cech covers.

The point is that it seems to me a folk statement that each simplicial presheaf which is a coproduct of representables is cofibrant. But I don't know any reference for that!!! The only results that I know is the one of Dugger


where there are some free-degenerancies conditions. So what about simplicial manifolds?


marked as duplicate by Dmitri Pavlov, Community Nov 6 '15 at 16:08

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  • 2
    $\begingroup$ The local projective model structure, being a left Bousfield localisation, has the same cofibrations as the projective model structure. In particular, the cofibrant objects are the same. The representables are cofibrant in the projective model structure, almost by definition. However, a simplicial presheaf that is degreewise representable may not be cofibrant – that is what the free degeneracies condition is about. $\endgroup$ – Zhen Lin Nov 5 '15 at 18:17
  • $\begingroup$ A necessary and sufficient condition for projective cofibrancy is given here: mathoverflow.net/questions/97690/… $\endgroup$ – Dmitri Pavlov Nov 6 '15 at 10:39