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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?

1) Is it true that the the cartesian product between cofibrant objects is again cofibrant?

2) If not, what about the cartesian product between a cofibrant object and a representable simplicial presheaf?

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  • $\begingroup$ So, you are assuming $\mathcal{C}$ has products? $\endgroup$ – Zhen Lin Feb 20 '15 at 12:04
  • $\begingroup$ Yes $\mathcal{C}$ has products. $\endgroup$ – Cepu Feb 20 '15 at 13:32
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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.

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  • $\begingroup$ Thanks! I have another question in mind. Assume that $\mathcal{D}$ is another small category, then consider the cartesian product $\times \: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{D})\to sPsh(\mathcal{C}\times \mathcal{D})$. Then is it true that the product of two cofibrations is again a cofibration ? I think that your proof carry over this situation. $\endgroup$ – Cepu Mar 3 '15 at 14:19
  • $\begingroup$ Yes, it does seem to carry over. $\endgroup$ – Zhen Lin Mar 3 '15 at 17:28

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