Post date: 10-sep-2015 2:47:45
The purpose of this talk is to introduce the canonical model structure on Cat, also called folk model structure, whose weak equivalences are the equivalences of categories, whose cofibrations are the functors injective on objects, and fibrations the isofibrations. We show that this is a cofibrantly generated model structure, and induces a model structure restricted to Groupoids. In future talks we will be interested in homotopy colimits in this category since they are closely related to pseudo-colimits.
Then we show that there is a cofibrantly generated model structure on sSet due to Quillen, we follow a proof of Rick Jardine using countable trivial cofibrations as the generating trivial cofibrations. Another proofs are available, but they depend on a very deep and technical result by Quillen, that the geometric realization of a Kan fibration is a Serre fibration, which is also the key to show that sSet and Top are Quillen equivalent via the geometric realization-singular complex adjunction (see Homotopy Hypothesis).