Geometric realisation of smooth $\infty$-stacks

Question

Let $Sh^\infty(\mathsf{Man})$ denote the $\infty$-category of sheaves of $\infty$-groupoids over the site $\mathsf{Man}$ of smooth manifolds (if you prefer, that's the model category of simplicial sheaves on $\mathsf{Man}$), and let $\mathcal S$ denote the $\infty$-category of $\infty$-groupoids (the usual model category of simplicial sets).

The inclusion $\mathcal S\to Sh^\infty(\mathsf{Man})$ admits a left adjoint $$ \|\cdot\|:Sh^\infty(\mathsf{Man})\to\mathcal S $$ called geometric realisation.

Given two morphisms $f,g:X\to Y$ in $Sh^\infty(\mathsf{Man})$

let us write $f\sim g$ if there exists a (necessarily invertible) 2-morphism $f\Rightarrow g$ in $Sh^\infty(\mathsf{Man})$, and

let us write $f\approx g$ if there exists a map $h:X\times\mathbb R\to Y$ such that $h|_{X\times\{0\}}\sim f$ and $h|_{X\times\{1\}}\sim g$.

Is it true that for all $M\in\mathsf{Man}$, and all $X\in Sh^\infty(\mathsf{Man})$, the obvious map $$ \qquad\quad Hom_{Sh^\infty(\mathsf{Man})}(M,X)/\approx\quad \to \quad Hom_{\mathcal S}(\|M\|,\|X\|)/\sim\qquad\quad(*) $$ is bijective? [In the RHS of (*), the symbol ∼ just means "homotopic" (and there's only one notion of two morphisms in $\mathcal S$ being homotopic)]

What can be said about the class of objects $M\in Sh^\infty(\mathsf{Man})$ with the property that $\forall X\in Sh^\infty(\mathsf{Man})$ the map $(*)$ is bijective?

Answer 1

The case when $M$ is a smooth manifold follows from the smooth Oka principle. See there for an expository account of the argument and references to additional sources.

Indeed, the left side of (*) is $$\def\Hom{\mathop{\rm Hom}} π_0(ʃ\Hom(M,X)),$$ whereas the right side of (*) is $$π_0(\Hom(ʃM,ʃX)),$$ where $ʃ$ denotes the shape functor, which is called “geometric realization” in the main post and is denoted by $‖{-}‖$ there. (From my point of view, a geometric realization functor converts a categorical object like a simplicial set to a geometric object like a topological space, whereas a shape functor converts a geometric object like a sheaf of simplicial sets on manifolds to a categorical object like a simplicial set.)

Concerning the case of a general $M$, not much can be expected if $M$ has homotopy groups in degree 1 or higher (meaning $M$ is not weakly equivalent to a sheaf of sets on manifolds). For example, taking $M=N/\!/G$ to be the stacky quotient of a manifold by an action of a Lie group, and $X$ to be the stack given by the homotopy group completion of the sheaf of symmetric monoidal groupoids of vector bundles, the left side of (*) computes the equivariant K-theory of $M$ (basically, it boils down to Segal's model), whereas the right side computes the Borel equivariant K-theory of $M=N/\!/G$. These two are different in general.