# Function complex and simplicial presheaves

Let $\mathcal{C}$ be a small catgeory, $\mathcal{E}$ be a model category and $A\: : \: \mathcal{C}\to \mathcal{E}$ be a functor. Let $\tilde{A}$ be an objectwise cosimplicial frame on A. Consider the functor $\mathcal{E}\to sPsh(\mathcal{C})$ $$E\to \operatorname{Hom}_{\mathcal{E}}(\tilde{A}(-),E),$$ where $sPsh(\mathcal{C})$ is the category of simplicial presheaves equipped with the projective model structure. It is easy to show for example that if $E$ is fibrant then the above simplicial presheaf is fibrant. Do you know some conditions on $\mathcal{C}, \mathcal{E}, \tilde{A}$, $E$ such that the above simplicial presheaf $\operatorname{Hom}_{\mathcal{E}}(\tilde{A}(-),E)$ is cofibrant?

• What is B? Also, describing the context in which such a question arose would be helpful (and encouraged on MO). – Dmitri Pavlov Feb 13 '15 at 19:42
• Ups sorry $E$ not $B$!! – Cepu Feb 16 '15 at 7:21
• For a hom to be fibrant one normally expects its target (in this case E) to be fibrant, not cofibrant, so I don't understand the claim that Hom(A(−),E) is fibrant for cofibrant E. – Dmitri Pavlov Feb 16 '15 at 18:43
• Thanks and sorry again, actually the context is very general. – Cepu Feb 17 '15 at 16:37
• In such a general situation with no additional context it's unlikely that something better than the general cofibrancy criterion, as described in mathoverflow.net/a/127187, is possible. – Dmitri Pavlov Feb 24 '15 at 16:56