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@GregoryArone: In a setting without universes, Barwick's conditions simply state that a class is a set (in the usual sense). The Rezk–Schwede–Shipley construction works fine too.
The Čech cohomology mentioned in the second bullet point is defined as the Čech cohomology of any Lie groupoid that presents M, e.g., the one induced by an open cover of M. It doesn't have to be the trivial groupoid on M.
Partitions of unity are necessary to prove the second bullet point (higher cohomology of fine sheaves on paracompact Hausdorff spaces vanishes). If you impose it as a separate condition instead, there is no need to keep asking for fineness.
@JonPridham: However, in complete analogy to the Dold–Kan correspondence, the normalized chains functor does induce a Quillen equivalence from simplicial C^∞-rings to dg-C^∞-rings, as shown in Theorem 1.1 of arxiv.org/abs/2303.12699.
Here is a trivial observation about the functor $\sf O$: it sends all simplicial spheres $S^n=Δ^n/∂Δ^n$ ($n>0$) as well as $Δ^0$ to the same cosimplicial set. Thus, $\sf O$ destroys a lot (most?) information in a simplicial set.
@fosco: What model structure do you have in mind for cosimplicial sets? Are there any such model structures in the literature? Certainly, one has a nontrivial class of weak equivalences (natural transformations whose homotopy limits in the ∞-category of spaces are equivalences).
@მამუკაჯიბლაძე: I think the answer is negative already for 1-categories: there are many nonisomorphic braidings on a monoidal category that induce a commutative monoid structure on isomorphism classes of objects.
If A is braided (and not merely homotopy commutative), we can simply construct the delooping BA as an ∞-group and then take the category of spaces with an action of BA. Since BΩG≃G, this recovers the category of G-spaces. If A is merely homotopy commutative, I have an impression there is not enough data left to reconstruct G-spaces, since G can be reconstructed from the ∞-category of G-spaces, but G cannot be reconstructed from the ∞-group ΩG, unless we equip ΩG with a braiding. (Indeed, the ∞-group ΩG has exactly the data needed to reconstruct G as a space, not ∞-group.)
Murre proves that the resulting cocone over the indicated diagram of sets is a colimit cocone. In particular, his argument proves that the colimit exists.
