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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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  • $\begingroup$ 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. $\endgroup$ – მამუკა ჯიბლაძე Jul 10 '14 at 16:42
  • $\begingroup$ 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). $\endgroup$ – David Carchedi Jul 10 '14 at 16:58
  • $\begingroup$ In my work I have encountered several situations that precisely match your description: the derived mapping space can be computed as the strict mapping space. However, in many cases this was the result of an indirect computation, i.e., I computed both the spaces and then discovered that they are equal. In some situations of this type it was convenient to bypass model structures at some stage and use other tricks (e.g., homotopy (co)limits etc.). $\endgroup$ – Dmitri Pavlov Jul 31 '17 at 9:08

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