Hammock localization in Waldhausen categories Original Research Article

Let image be a Waldhausen category, alias category with cofibrations and weak equivalences. The hammock localization (Dwyer and Kan, 1980) of image with respect to the subcategory cof image is a category containing image, with the same objects as image, where for each D and E the morphisms from D to E form a simplicial set morH(D, E). Morphisms in cof image become invertible up to homotopy in the hammock localization. The main result states, with mild hypotheses on image, that for any E in image the contravariant functor D maps to morH(D, E) takes cofiber squares to homotopy pullback squares.