Constructions of Factorization Systems in Categories

In [2] we constructed homological localizations of spaces, groups, and 17"modules; here we generalize those constructions to give "factorization systems" and "homotopy factorization systems" for maps in categories. In Section 2 we recall the definition and basic properties of factorization systems, and in Section 3 we give our first existence theorem (3.1)for such systems. It can be viewed as a generalization of Deleanu's existence theorem [5] for localizations, and is best possible although it involves a hard-to-verify "solution set" condition. In Section 4 we give a second existence theorem (4.1) which is more specialized than the first, but is often easier to apply since it avoids the "solution set" condition. In Section 5 we use our existence theorems to construct various examples of factorization systems, and we consider the associated (co)localizations. As special cases, we obtain the Stone-Cech compactification for topological spaces, the homological localizations of groups and w-modules [2, Section 5], the Extcompletions for abelian groups [4, p. 171], and many new (co)localizations. I n Section 6 we generalize the theory of factorization systems to the context of homotopical algebra [9]. Among the "homotopy factorization systems" in the category of simplicial sets are the Mo0re-Postnikov systems and the homological factorization systems of [2, Appendix]. In Section 7 we generalize 4.1 to give an existence theorem for homotopy factorization systems. This leads to "Andersonlike" localizations (7.3) and p-completions (7.4) in the pointed simplicial or CW homotopy category. It also leads to "colocalizations of spaces with respect to homotopy theories" (see 7.5). We will use the language of Godel-Bernays set theory, distinguishing between "sets" and "classes". The objects of a category C will form a class, but C(X, Y) is "required to be a set for each X, Y ~ C.