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We construct three simplicial presheaves on the site of ringed spaces, and inparticular on that of complex manifolds. The descent objects for thesesimplicial presheaves yield Toledo--Tong's twisting cochains, simplicialtwisting cochains, and complexes that appear in Green's thesis on Chern classesfor coherent analytic sheaves, respectively. We thus extend the aforementionedconstructions to the equivariant setting, and more generally to stacks. This isthe first step in achieving push-forwards in K-theory and Riemann--Rochtheorems for appropriate stacks, as was achieved by Toledo and Tong forarbitrary complex manifolds, and further pursued by O'Brian and Green.

Simplicial presheaves of Green complexes and twisting cochains