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Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on $\mathbf{Set}_\Delta^{\mathscr{C}^{op}},$ (most notably the injective and projective model structures) which present the $\left(\infty,1\right)$-category of functors $$\mathbf{Fun}\left(\mathscr{C}^{op},\infty\mbox{-}\mathbf{Gpd}\right)$$ (i.e. $\infty$-presheaves). $Y$ can be identified with a $0$-truncated object therein, and associated to $X_\bullet$ we can also choose an object $\mathscr{X}$ which is well defined up to equivalence.

Question Under what conditions (which are checkable in practice!) on $X_\bullet$ is $$\mathbf{Map}(Y,\mathscr{X})$$ weakly equivalent to the simplicial set whose $n$-morphisms are $\mathbf{Hom}\left(Y,X_n\right)?$ (The latter is the naive simplicial mapping space coming from the simplicial enrichment of $\mathbf{Set}_\Delta^{\mathscr{C}^{op}}.$)

I don't want to assume that $Y$ is a coproduct of representables (so probably assuming $Y$ is projectively cofibrant is not reasonable), but also I don't want to assume that $X_\bullet$ is injectively fibrant, since I have no idea how to check if this is true (what are the generating trivial cofibrations??). I'm hoping that there is perhaps an intermediate model structure which could be useful. I don't want to make any assumptions on $\mathscr{C}$ (like it being Reedy).

Note: In the situation I care about, I have a fully and faithful functor $$k:\mathscr{C} \hookrightarrow \mathscr{D}$$ and $Y=k^*y(d),$ where $y(d)$ is a representable presheaf (of sets) on $\mathscr{D}.$

So, what could be useful would be a model structure (compatible with the standard enrichment in simplicial sets and presenting the functor $\infty$-category) in which each $Y=k^*y(d)$ is cofibrant and in which there are checkable conditions to see if an object is fibrant. Of course, I welcome any answer to this question, even if it doesn't use model categories.

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In which category do you take Map? If just in simplicial presheaves, then I do not quite understand what do model structures have to do with it. If not, I do not understand precisely which universal property does characterize this Map. – მამუკა ჯიბლაძე Jul 10 '14 at 16:42
I am taking "Map" in the $\left(\infty,1\right)$-category $\mathbf{Fun}\left(\mathscr{C}^{op},\infty\mbox{-}\mathbf{Gpd}\right)$, or equivalently, I am taking the derived mapping space in (any of the equivalent) model category(ies). – David Carchedi Jul 10 '14 at 16:58

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