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One relevant paper is arxiv.org/abs/0810.4935v1, which constructs differential K-theory from vector bundles with connection using a variant of group completion.
As far as I can see, there is no question being asked here. Also, what exactly is the space S^∞? It would be best to give references for the source you are using.
@RonaldJ.Zallman: The functor B commutes with homotopy products. Lifting abelian group objects in Ho(CW) to E_∞-spaces is a nontrivial problem, the space of liftings is in general noncontractible. I am not aware of any general theorems, perhaps more could be said for a specific problem.
@RonaldJ.Zallman: In order to define the delooping functor B, it is not sufficient to have an abelian group object in Ho(CW). Way too much information is lost when passing to the homotopy category, and in general, in the modern literature you no longer see (for the most part) people working with algebraic constructions in Ho(CW) in the first place. To define B, you need a (homotopy coherent) abelian group object (more generally, an abelian monoid object) in the actual category of spaces, not in the homotopy category.
@JackYo: Maps into BG^S and BG classify different things. The derived space of maps of spaces X→BG^S is equivalent to the ∞-groupoid of principal G-bundles over X, their concordances, and higher concordances (parametrized by the simplex Δ^k) as k-morphisms. The derived space of maps RMap(X,BG) of ∞-sheaves (and not just spaces) X→BG is equivalent to the groupoid of principal G-bundles over X and their isomorphisms. Finally, the ∞-groupoid RMap(X,BG^S) is equivalent to the shape of the derived internal hom RHom(X,BG), whereas RMap(X,BG) is equivalent to RHom(X,BG)(R^0).
@JackYo: Spaces could be taken to mean simplicial sets here. Groupoids embed into simplicial sets via the nerve functor. BG is a presheaf of groupoids (and therefore simplicial sets) on the site of cartesian spaces. (The space BG used in the first two paragraphs of my answer is not the same BG, but rather the shape of the presheaf BG.) Thus, the derived mapping object from X to BG is itself naturally a groupoid, the groupoid of principal G-bundles over X.
@JackYo: Yes, that's correct. The groupoid of principal G-bundles over X is the dervied mapping space (or groupoid) RMap(X,BG). RMap can be computed as the right derived functor of Map (the simplicial mapping space), by cofibrantly replacing X and fibrantly replacing BG. If BG is an ∞-sheaf, then BG is fibrant in the local projective model structure on simplicial presheaves. A cofibrant resolution of X is given by the Čech nerve of a good open cover of X. Altogether, we get the groupoid of maps from the Čech nerve to BG. Thus, Čech cohomology computes sheaf cohomology in this context.
@JackYo: There is no equivalence relation used in the construction. I also added a paragraph about the other approach, where the answer is positive at least in some cases.
What exactly is the relevance of the Kan–Thurston theorem here? The OP explicitly states that X is a classifying space for a discrete group G, which by definition makes it weakly equivalent (and not merely homology equivalent) to K(G,1).
@GeometriaDifferenziale: Dividing by exp(…) makes the resulting section smooth and bounded in norm. Then multiplying by h makes the resulting section extendable by zero to a section over M. If we did not divide by exp(…), the resulting quotient could go to infinity as we approach the boundary of U.
@DamienC: Flabby means the restriction morphism F(V)→F(U) is surjective for any inclusion of opens U⊂V. If F is the sheaf of sections of the trivial vector bundle, it is easy to construct examples where the restriction map is not surjective, e.g., the smooth section x↦1/x over (-∞,0)∪(0,∞) does not extend to the real line.
