Post date: 20-jul-2015 14:05:28
We introduce the basic theory of weak factorization systems developed by A. Joyal, and within this theory we introduce (Quillen's closed) model categories. We recall the not-so-well-know observation of A. Joyal that weak equivalences are automatically closed under retracts in a (closed) model category. We start a first approach to some examples of model categories but putting off the proof of the axioms. We also introduce Quillen functors and Quillen adjunctions. Then we prove Ken Brown's lemma, and as a corollary, that every Left Quillen functor is homotopical between cofibrant objects. This result combined with the small object argument will show that Left Quillen functors can be derived by applying (a functorial) cofibrant replacement.