What is the appropriate $(\infty,1)$-site structure on SmoothMfld, making the de Rham complex a $D(\mathbb{R})$-valued $(\infty,1)$-sheaf?

Question

Let Mfld denote the category of smooth manifolds. This also has an $(\infty,1)$-category structure, with higher cells being smooth n-fold homotopies.

We know Mfld has a 1-site structure / 1-Grothendieck topology, generated from the Grothendieck pretopology whose covering families into $X$ are families of maps $\{f_i : U_i \rightarrow X\}$ such that each $f_i$ is a smooth open embedding, and the unions of the images of the $f_i$ is $X$.

For $R$ a commutative ring, let $D(R)$ denote its derived $(\infty,1)$-category, the $(\infty,1)$-localization of the chain complex category $\mathrm{Ch}R$ at the quasi-isomorphisms.

I am reading the book Differential Cohomology, and in Example 3.3.4 the authors say that we can view the de Rham complex $\Omega^{\bullet}(-)$ as a $D(\mathbb{R})$-valued sheaf on Mfld. I assume this is in fact an $(\infty,1)$-sheaf with respect to some $(\infty,1)$-site structure on Mfld?

However, the authors only describe (in 3.1.1) the 1-site structure on Mfld, not an $(\infty,1)$-site structure. I was wondering, is there any "natural" choice of $(\infty,1)$-Grothendieck topology on Mfld in this situation?

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In particular, I was wondering if it is possible to say there is some $(\infty,1)$-site structure on Mfld, such that $D(\mathbb{R})$-valued $(\infty,1)$-sheaves on Mfld are in bijection with 1-sheaves, valued in the homotopy category $\mathrm{h}D(\mathbb{R})$, perhaps satisfying some property involving the model structure on $\mathrm{h}D(\mathbb{R})$?

Answer 1

However, the authors only describe (in 3.1.1) the 1-site structure on Mfld, not an (∞,1)-site structure. I was wondering, is there any "natural" choice of (∞,1)-Grothendieck topology on Mfld in this situation?

An (∞,1)-site structure on a 1-truncated ∞-category (i.e., an ∞-category equivalent to an ordinary 1-category $C$) is the same thing as an ordinary site structure on $C$. So the answer is yes, and it is the (∞,1)-site structure canonically induced from the 1-site structure.

In general, a Grothendieck (∞,1)-topology on an (∞,1)-category is just an ordinary Grothendieck topology on its homotopy category, as already observed by Toën and Vezzosi.

In particular, I was wondering if it is possible to say there is some (∞,1)-site structure on Mfld, such that D(R)-valued (∞,1)-sheaves on Mfld are in bijection with 1-sheaves, valued in the homotopy category hD(R), perhaps satisfying some property involving the model structure on hD(R)?

No. Passing to the homotopy category destroys most of the information about (derived) mapping spaces, so there is no chance to establish a bijection between the fully homotopy coherent notion of an ∞-sheaf on Mfld to 1-sheaves valued in the homotopy category, even if we take isomorphism classes on both sides.

This also has an (∞,1)-category structure, with higher cells being smooth n-fold homotopies.

Sheaves on this (∞,1)-category (with the natural Grothendieck topology) are equivalent to sheaves on the (∞,1)-category of spaces (alias simplicial sets). So the resulting notion of an (∞,1)-site is not very interesting.