Duality for categories of presheaves

I will give examples of small categories for which there is a bar-cobar duality between the category of presheaves of chain complexes on that small category and the category of copresheaves of such. Several up-to-homotopy (co-)algebraic structures can be defined terms of such (co-)presheaves equipped with extra structure, and this extra structure is respected by the duality. The duality specializes to bar-cobar duality theorems for infinity-algebras, infinity-operads and algebras over the latter, and equivariant versions of these.