Hammock localization in Waldhausen categories
A Waldhausen category \({\mathcal C}\) is a category with cofibrations and weak equivalences satisfying certain axioms. It is the proper setting to construct and study algebraic K-theory [see \textit{F. Waldhausen}, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)]. The study of the algebraic K-theory of a Waldhausen category \({\mathcal C}\) leads to Dwyer-Kan's hammock localization [\textit{W. G. Dwyer} and \textit{D. M. Kan}, J. Pure Appl. Algebra 18, 17-35 (1980; Zbl 0485.18013)] \(L^H({\mathcal C})\) of the category \({\mathcal C}\) with respect to its full subcategory in which morphisms are cofibrations and weak equivalences. Denote by \(\text{mor}^H(D,E)\) the simplicial set of morphisms in \(L^H({\mathcal C})\) between any two objects \(D,E\). The main result in this paper is theorem 2.1 stating that, under certain conditions, if the second variable is fixed, the contravariant functor \(\text{mor}^H(-,E)\) takes cofiber squares to homotopy pullback squares.
scientific article; zbMATH DE number 1308367