Factorization Systems

These notes were written to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in Fall 2008 at the University of Chicago. We first introduce orthogonal factorization systems, give a few examples, and prove some basic theorems. Next, we turn to weak factorization systems, which play an important role in the theory of model categories, a connection which we make explicit. We discuss what it means for a weak factorization system to be functorial and observe that functoriality does not guarantee the existence of natural lifts. This leads us, naturally one might say, to the definition of a natural weak factorization system, which is where we conclude these notes. The reader is assumed to have some familiarity with category theory — functors, limits and colimits, naturality, monads and comonads, comonoids, 2-categories, and some basic categorical terminology; [9] is a good reference for any concepts that may be unfamiliar. 1. Orthogonal Factorization Systems Definition 1.1. An orthogonal factorization system in a category K is a pair (L,R) of distinguished class of morphisms such that (I) L and R contain all isomorphisms and are closed under composition. (II) Every f ∈ mor K can be factored as f = me with e ∈ L and m ∈ R. (III) This factorization is functorial, i.e., given the solid diagram (1.2) · u //