Post date: 03-ago-2015 16:01:32
As we discussed in the previous talk, it's a difficult task to show that a category (with three distinguished classes of arrows) form a model category. We approach to this problem by first solving an easier one: how to show that a pair of classes of arrows (L,R) form a weak factorization system. A general procedure to construct a wfs is to start with a class of arrows I, then take R=rlp(I), and L=llp(R), this pair automatically satisfies all except the factorization axiom i.e. that for every arrow there is a factorization by an arrow of L followed by one of R. By the previous talk, we are also interested in some sort of "functoriality" of the factorization, needed in order to construct derived functors. We will show that Quillen's small object argument produce a functorial weak factorization system at the cost of some technical smallness condition. This combined with the previous results finish the proof that Left Quillen functors can be derived by applying a functorial cofibrant replacement.
Finally, we introduce the notion of cofibrantly generated model categories and as an upshot, we get a "recognition theorem" that makes a significative reduction of stuff to check in order to prove that some data defines a model category structure.