  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.
