The Bousfield localizations and colocalizations of the discrete model structure (Q511053)

In this article, the author answers the following questions about model categories [\textit{D. G. Quillen}, Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903); \textit{M. Hovey}, Model categories. Providence, RI: American Mathematical Society (1999; Zbl 0909.55001)]. (1) If \(\mathcal{C}\) is a category and \(\mathcal{A}\) is a reflective subcategory of \(\mathcal{C}\), ``is there some model structure on \(\mathcal{C}\) in which \(\mathcal{A}\) is the subcategory of fibrant objects, and such that the reflector functor is a fibrant replacement?'' (2) ``Given an idempotent monad on a category \(\mathcal{C}\), is there some model structure on \(\mathcal{C}\) such that the monad is a fibrant replacement monad?'' (3) ``What are all the Bousfield localizations and Bousfield colocalizations [\textit{A. Bousfield}, Topology 14, 133--150 (1975; Zbl 0309.55013); \textit{P. Hirschhorn}, Model categories and their localizations. Providence, RI: American Mathematical Society (2003; Zbl 1017.55001)] of the discrete model structure?'' (4) ``What is the homotopy category of each of those Bousfield localizations and Bousfield colocalizations?'' (5) ``What are the algebraic \(K\)-groups [\textit{F. Waldhausen}, Lect. Notes Math. 1126, 318--419 (1985; Zbl 0579.18006)] of each of those Bousfield localizations and Bousfield colocalizations?'' (6) ``Which categories are equivalent to the homotopy category of some Bousfield localization or Bousfield colocalization of the discrete model structure?''