Simplicial presheaves of Green complexes and twisting cochains

We construct three simplicial presheaves on the site of ringed spaces, and in particular on that of complex manifolds. The descent objects for these simplicial presheaves yield Toledo–Tong’s twisting cochains, simplicial twisting cochains, and complexes that appear in Green’s thesis on Chern classes for coherent analytic sheaves, respectively. We thus extend the aforementioned constructions to the equivariant setting, and more generally to stacks. This is the first step in achieving push-forwards in K-theory and Riemann–Roch theorems for appropriate stacks, as was achieved by Toledo and Tong for arbitrary complex manifolds, and further pursued by O’Brian and Green.