Let $Psh(\mathcal{C})$ be the category of simplicial presheaves equipped with the projective model structure. The cartesian product between two representables presheaves is clearly again representable and hence cofibrant, what about a more general statament?



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> 1) Is it true that the the cartesian product between cofibrant objects is again cofibrant?
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> 2) If not, what about the cartesian product between a cofibrant object and a representable simplicial presheaf?
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Cartesian products between cofibrant simplicial presheaves
If $\mathcal{C}$ has finite products, then the class of projective cofibrations is also closed under finite products. Indeed, since the cartesian product in the category of simplicial presheaves preserves colimits in each variable, it suffices to check the claim on the generating projective cofibrations, i.e. that for every pair $(A, B)$ of objects in $\mathcal{C}$ and every pair $(n, m)$ of natural numbers, the inclusion
$$(\partial \Delta^n \odot h\_A) \times (\partial \Delta^m \odot h\_B) \hookrightarrow (\Delta^n \odot h\_A) \times (\Delta^m \odot h\_B)$$
is a projective cofibration. But this is isomorphic to the inclusion
$$(\partial \Delta^n \times \partial \Delta^m) \odot h\_{A \times B} \hookrightarrow (\Delta^n \times \Delta^m) \odot h\_{A \times B}$$
and $\partial \Delta^n \times \partial \Delta^m \to \Delta^n \times \Delta^m$ is a cofibration of simplicial sets, so the claim follows from the fact that $(-) \odot h\_{A \times B}$ sends cofibrations of simplicial sets to projective cofibrations of simplicial presheaves.


A similar argument works for projective trivial cofibrations.


